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This is a great question that unfortunately falls victim to the classic difficulty of generalizing the medieval period: it all depends. Individual journeys could be made for many reasons with many different elements, from the wealth of the individual to the purpose of the journey to the resources available at the journey's origin, at its destination, and along its travel route. So before we get into some details you should know that a traveler might, sometimes and for some reason, go Gandalf-style more or less alone, or join a huge train of merchants and carters, or travel armed with an armed entourage. One might also stay in high style at inns along the way, or sleep in roadside ditches or makeshift camps, and take some succor at monasteries or religious hostels. One person on a single journey might, at some time or another, do all of these things, as well as book passage on boats for river, lake, or sea travel.

To answer this I'm going to break a hypothetical journey into phases, and use examples from a variety of sources to explain some of the choices a hypothetical traveler might have before them. I'll try to keep it relatively close to the HRE in the 13th century, but by necessity some examples will be drawn from other places and other periods.

Setting Out

Upon setting out, however, the only help I received from my abbot was a single horse. Lacking in money, a change of clothes, and other necessities, I arrived in Orbais, a place known for its great hospitality. There I was refreshed by the conversation of the lord abbot D and sustained by his generosity, and on the next day I undertook to travel as far as Meaux.

This was written by a monk, Richer, from the French monastery St. Reimi in 991. He had just received a letter brought to him by a single rider - a knight - sent by a colleague from Chartres, who wished to speak with him. Richer "bestowed a kiss" on the knight, and set to travel with the man to Chartres, adding to the small party a boy, presumably a member of Richer's clerical order to act as a servant. It's clear from the tone that Richer, who may not be the most experienced traveler, was somewhat disappointed by the assistance offered by his abbot. He was given no money and only a single horse and servant, and he knows that the journey ahead will likely be a difficult one.

Travel

Richer is unspecific about the time all of this takes, but it's likely that the Orbais he referred to is Orbais-l'Abbaye in modern France, about 60km southwest of St. Reimi, near Reims. He points out that he took rest and was refreshed there, and it was likely that nearly any abbey or monastic hostel would accept a traveling monk in a similar way, though Richer points out that Orbais was known for it beforehand; so perhaps it was not so common after all, and travelers had to rely on prior knowledge of which places would succor weary travelers and which wouldn't - and even if they were of a subculture that would be accepted. If Richer was a traveling merchant, would the abott of Orbais treat him so well? After arriving there, his horse - formerly a Bucephalus - suddenly loses its ardor, and then they wander six leagues out of their way, after which the horse died and they were beset by a terrific storm. Then the boy, utterly exhausted, laid down by the side of the road.

"Those who have ever suffered similar misfortunes can judge from their own experiences how great my agitation and anxiety were at that moment."

Richer leaves the boy with the baggage and makes his way with the faithful knight to nearby Meaux, where he can't cross the bridge because the planks are riddled with holes. The knight tries to find a boat but fails, so they have to try to cross the bridge, dangerous as it is. Less so for them than for the horses:

Sometimes putting a shield down under the horses’ feet in the gaping holes and sometimes joining together discarded planks, sometimes bending down and sometimes standing up straight, sometimes coming forward and sometimes running back, he successfully made it all the way across the bridge with the horses, while I accompanied him.

This is a comically difficult journey, I want to be clear. Not every traveler would have an avalanche of difficulties hit them at very stage. In fact some journeys are so unremarkable that their authors breeze over them completely. One such, drawn from an extensive book entitled BOOK OF THE KNOWLEDGE OF ALL KINGDOMS, LANDS, AND LORDSHIPS THAT ARE IN THE WORLD, describes the author's journey through a series of HRE cities in a rather breezy manner:

From there I went along the coast to a city which they call Zeeland and from there to another called Maxa , and to another Lübeck which are cities of Germany. From there I passed to the great and rich city of Dordrecht, crossing a great river which they call the Rhine, having its origin in the German Alps. This river passes by Cologne, a great city of Germany. In this city they say that the three Magi Kings are interred, who worshipped Jesus Christ in Bethlehem. But when I traveled in the empire of Cathay [northern China] I was in a city called Solin and they showed me three highly revered monuments and they were in honor of the three Magi Kings who adored Jesus Christ, and they said that they were natives of that city.

Here's half a dozen cities of the empire passed by, a throwaway reference to the author's visit to China, and the rest of the book describes his travel through places as far away as Norway and Africa, even including Gog and Magog, given by the author as in northern China and inching toward Mongolia.

For religious travelers - proper pilgrims, not those traveling on what we might consider business as Richer was - one's journey might be mapped out on a traveler's itinerary. This was a sort of map, in the same way that a ship's rutterbook was a map. Not a geographical abstraction of the land itself - those types of maps remained symbolically abstract until more or less the end of the 15th century - but a list of places with travel tips connecting them, written down in a simple and chronological manner. An example is the Einsiedeln Itinerary, focusing on the city of Rome. It appears almost like a traveler's guide today, with attention paid to famous landmarks and tips on getting around the city. The portion that covers leaving the city is most relevant here:

Roads outside the city

On the Portensian Road outside the city on the right. Church of Sts-Abdon and Sennen.

On the Aurelian Road outside the city on the right. Church of Sts-Pancras, Processus, and Martinian.

On the Salerian Road outside the city on the right. Church of St-Saturninus, Church of St-Felicity and her Seven Sons.

On the Pincian Road outside the city on the right. Church of St-Basilissa, Church of St-Pamphilius. Church of Sts-Protus and Hyacinth. Church of St-Hermes, Church of the Head of St-John.

Provided you knew - or could ask a local about - these landmarks, getting around could be relatively painless. But then, knowing local languages and customs, or having knowledge about places in which travelers from your country could be found, would be intensely important, and thus having access to experience travelers or guides was an important factor in a successful journey. Hired guides were a reliable element of a lot of far-flung destinations, but were also a resource for humble inexperienced travelers like Richer. This was also, however, an opportunity for robbery and pillage, with false guides luring the unwary into ambushes or simply leading them to an unknown area and robbing them.

Those travelling on business seldom mentions the details of travel unless they were remarkable; a series of misfortunes, a robbery, or some other setback. Most are concerned with their business, such as the state of their apprenticeship, or the accounts information on goods sold or retained. The 16th century apprentice Michael Behaim wrote home to his guardian informing him that he was displeased with his new station, and that his master used him for sweeping and other labor, and neglected his education in the business he hoped to pursue. The journey he had taken, from Nuremberg to Milan at the age of twelve, is undescribed. Nevertheless, frequent letters from Milan to Nuremberg flew back and forth, implying that there was a relatively stable course of travel between the two cities that made communication reliable. Michael does, at one point, request that his guardian (his cousin Friedrich) write to another relative, Gabriel Futterer, to furnish him with a horse so that Michael could accompany his master on an Easter pilgrimage to the Shrine of St Mary at Loredo, a journey of five German miles (about 3-5 modern miles each) from Rome. Quite a distance for a holiday pilgrimage! But Michael's request also reveals some anxiety:

If I do not go with him now, I will have to go at another time. But this is the best time for me to go. I may then carry out my vow and see and learn something along the way about commerce, so that when I come home I will be both a man and my own man. Should I be abroad for four years and see no city but Milan, and learn nothing, I will be a laughingstock.

tl; dr: Frankly speaking, almost every aspects of OP's premise suggests that he must have been very exceptional figure in Lübeck around 1200 CE, so it would be very difficult to generalize into HRE.

1: .....Just a well of peasant-merchant.

Unfortunately, we have little extant evidence on the social origin of early citizen/ merchants as well as early historical development of the city Lübeck after its alleged foundation by Henry the Lion in 1158/59 to the late 1220s (when Frederick II granted the privilege as an (free) Imperial city). A recently popular (though perhaps a bit controversial) hypothesis is, however, that former Ministerialität (ministeriales), lesser nobility or knights of possibly legally ex-unfree origin but with some financial capitals, might have played an much more important role in Lübeck's early history (Jahnke 2009: 55-58). In short, the core part of the early merchants might have been drawn from the lesser nobles instead of peasants. Neither is it so clear as previously assumed that the political as well as economic dominance became stable by the end of the 12th century around the southern Baltic (the collapse of Danish 'Baltic' empire in the 1220s enabled Lübeck to expand their influence further at first).

Based on these circumstanced, to be armed by yourself might not be so common, but OP 'could' (or de facto 'was forced to') armed bodyguard along the route in exchange for the toll tax anyway, as we'll see below.

2: how a secular traveler; not a pilgrim

What kind of business a secular traveler (merchant) had in Rome around 1200?

AFAIK the first local evidence of rising interest among possibly non-political travelers to Rome in the neighborhood area dates only back to the middle of the 13th century, not earlier and it is found in the monastic annals of Stade, a city in northern Germany (Saxony), though it discusses possible pilgrim routes from Saxony to Rome (Esch 2009: 19).

It was not until 1424 that the first German-Italian 'travel dictionary' was written by a Nürnberger and published in Venice. It includes many expressions useful primarily for merchants and the appearance of such kind of text presupposes the more frequent communication between Germany and Italy, including merchants in the eve of the Renaissance (Dürer also published some of the scenery on the roads from Germany to Italy, as well as a ground plan of rooms and their furniture in the hotel in Venice he stayed in his 2nd Italian travel in 1505/06!) (Esch 2009: 14-16).

Alternatively, if OP just needs Italian or other Mediterranean exotic goods north to alps, I would instead recommend him to take a visit in Champagne fairs in now Eastern France around Troys and Provins where caravans of the Italian merchants and their local agents, together with merchants from different parts of north-western Europe, gathered there (Cf. Face 1958). Even to the Champagne fairs, however, the joint venture of northern German merchants was not so common around 1200, however. Caravans of Italian merchants came to the fairs (some Champagne cities held the regular fairs in order within the year) by way of Alpine passes that were firstly open in spring.

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OK, the following is the information OP mainly want to know, I suppose.
What I summarize here is, however, mainly based on the some retrospect deduction from the 14th century sources.

The early 13th century law text, Saxon Mirror (Sachenspiegel), distinguishes the 'royal or impedial roads (Königs- oder Kaiserstrasse) ' from other ordinary roads. The (re-)discovery of Roman Law in the 12th century played an important role in securing this concept, and everal rulers across Latin West, including Frederick (I) Barbarossa and Frederick II of the Staufens, now regarded the public maintenance of such 'royal roads' as well as bridges in exchange for the collection of toll tax as part of their royal rights (regalia). It also includes a kind of primitive 'priority code for the travelers', and it firstly specifies the passage of wagons on the royal roads with an possibly expanded width (ca. 4 meters and often paved with stones) enough for two wagon to go at the same time (Spufford 2002: 180). What we are not so sure is, however. whether this kind of maintenance had already become norm across HRE where Author Eike of Repgow wrote Saxon Mirror in 1220s.

The 'maintenance' of good order of the road also included the safety of the passengers, and rulers (and some local authorities they delegated this duty) also made it rule to demand the fee of safe conduct/ passage (conductus in Latin; geleit in vernacular German) in exchange of providing armed escorts to the caravan as a part of this duty. You couldn't trust these bodyguards too much, since they sometimes also asked for some extra payments for their food, or even expected some kick backs from the inn along the road networks. Some German cities like Nürnberg also made their own special agreements with neighboring powers on this safe passage toll, and 'the book of safe passage toll' (Geleitbuch) from 15th century Nürnberg illustrates individual discounted price of the tolls as well as 'recommended' routes.

On the other hand, city states in northern Italy took it up as their own duty. Spufford illustrates the efforts of Pisa and Florence to expand the road networks on their suburban territory (contado)that [two] wheeled wagons could pass in course of the 13th century (Spufford 2002: 182f.).

Before such a maintenance became common, however, we should not presuppose that even all the major roads could have been passed with wagons easily (as even Gandalf enjoyed touring in the Shire). In such a condition, packed horse (with its porter) would be only the reliable means to transport. They could be hired alongside the route (see below), but it was not so [corrects]: in-expensive.

As for the main traveling routes and lodgings (inns), I suppose this 'Rome-route' map by Erhard Etlaub of Nürnberg (ca. 1500) could be used as the departure point for our considerations. There were several roadways across HRE in this 'south-up' map, and many of the cities depicted on the roadways had already been founded by the end of the 12th century. One dot in the road also roughly corresponds with the 7.4 kilometers, and Spufford seems to be confidant about the existence of inn in small settlements between the cities across HRE......at least in the 14th century, based on the 14th century source, the itinerary of Brugge that depicts inns along the roads at about 11-14 kilometers intervals (Spufford 2002: 207).

A famous 14th century Italian handbook of trade (Peglotti) also narrates the importance of such inns both in cities and out of cities along the roads. They offered not only the room and the food to the travelers, but also functioned as a point of meeting with new hired local carriers and their packed horse and wagons. The public warehouse was sometimes also attached to such a inn (Spufford 2002: 203-205).

Thus, late medieval merchants were expected to lodge the inn, not the monastery and other kind of hospitals as pilgrims generally did, also to secure the transport of their goods, at least in the populated area. Again, however, we should keep in mind that we don't really have enough exact contemporary evidence on these topics from Germany around 1200. As for the price, all I can say is the the cost of transport occupies considerable part of the total sold price of exotic goods, especially brought by the land transport before the 14th century.

On the other hand, the situation in mountain passes out of HRE, such as Alpine route like St. Bernard pass on Via Francigena would be considerably different. On there passes where were closed during winter, the hospitals were open not only to the pilgrims, but also many kind of passengers including merchants. Innkeepers and the warden of the hospital on such mountain passes also were responsible for arranging the guides on the pass, called marroniers, together with packed mules also for the tired travelers.

References:

• Esch, Arnold. Wege nach Rom. München: C. H. Beck, 2004 (2003).
• Face, R. D. 'Techniques of Business in the Trade between the Fairs of Champagne and the South of Europe in the Twelfth and Thirteenth Centuries'. The Economic History Review 10, no. 3 (1958): 427–38. https://doi.org/10.2307/2591262.
• Jahnke, Carsten. 'The Influence of the Hanseatic League on the Cities in the North-Sea and Baltic-Sea Area - Some Reflections on the Triad "Trade- Cities - Hanseatic League'. In: Archaeology of Medieval Towns in the Baltic and North Sea Area, ed. Nils Engberg et al., pp. 51-63. Copenhagen: National Museum of Denmark, 2009. PNM 17.
• Selzer, Stephan. Die mittelalterliche Hanse. Darmstadt: WBG, 2010.
• Spufford, Peter. Power & Profit: The Merchant in Medieval Europe. London: Thames & Hudson, 2002.

A follow-on question if I may. What would travellers have done for money? Would they really set off with enough money in their pocket for them to live off for weeks on end, and wouldn't that be a massive robbery risk? I presume the properly wealthy would have luggage and goods sent ahead, but our regular peasant-merchant wouldn't have access to anything like that?

My mind boggles at the mechanics of travel in this period, even moreso for the longer journeys/pilgrimages that would have taken months to complete.