Peter the Great constantly moved about Russia during his reign, checking on all sorts of different governmental projects and what-not. During this time Peter frequently read letters from provincial governors, familial correspondence, and classical literature. Catherine the Great, while not travelling nearly as much as Peter the Great did, read lots of early modern philosophy and literature. She even maintained correspondence with Voltaire, a famous French philosopher.

Peter the Great: His Life and World by Robert K. Massie

Catherine the Great: Portrait of a Woman by Robert K. Massie

I know this isn't the precise time period you're inquiring about, but it might be helpful as it's exactly the kind of travel you're describing through the lens of a primary source.

Laura Ingalls Wilder is most famous for the Little House series of books about the 19th century American West, but some of her letters and journals have also been collected and published. One such journal was published under the name On the Way Home and concerns the Wilder family's journey from De Smet, South Dakota to Mansfield, Missouri in 1894. There had been seven years of very little rain, the crops had failed, the banks had gone under, and Laura and Almanzo (her husband) were both struck by diphtheria. They made the decision to leave South Dakota and purchase a farm in the Missouri Ozarks. The property that became Rocky Ridge Farm is still around today, and the house is now a museum of Wilder's life and work.

Anyway! While this was long past the point that railroads had made their way to the West, the Wilders had to move their entire household, and few to no trains were running in that portion of the country due to the rampages of Coxey's Army that year. They packed everything into a covered wagon and left De Smet on July 17th, arriving in Mansfield on August 30th of the same year. It's described in the book as being a 650-mile journey; having just looked at it on Google Maps, the trip south on I-29 via car clocks in at 670 miles, so I believe this is accurate. The route given in the journal is actually fairly close to what Google Maps would tell you to do today, although the Wilders stayed more to the west and south, passing through Lincoln, NE and eastern Kansas rather than entering Missouri early.

As an interesting aside, the Wilders' journey took 44 days (although they did not travel on Sundays). The modern route via car will run you 10-12 hours if you drive it straight.

So what did they do for those 44 days? As you might guess, they were probably bored a lot -- Laura commented on the monotony of the journey -- but her notes do give a lot of insight into the things they were disposed to notice:

• Temperature: It was extremely hot; most days the temperature ranged from the early 90s to 110 F (34.4 to 43.4 C). Laura does grumpily record that she lost the thermometer on Sunday, August 5th.
• Neighbors: The traveling party (the Wilders were going to Missouri with another family, the Cooleys) invariably sought the locals both on and along the road. They needed to stop to water the horses, buy feed for their horses and chickens, buy food for themselves, get Laura's revolver repaired, and they also exchanged a number of asbestos fire-mats that Almanzo had in return for the neighbors' many kindnesses. The only mildly unpleasant interactions that Laura notes were when a Russian lady gave them biscuits that she had carried inside her blouse, and when they met two drunk men who wanted to switch horses with them. They were traveling through an area of the country largely dominated by Russian and German immigrants who had done well in the West but were, like themselves, worried about the continuing drought.
• Food: The subject of what they ate came up regularly. Laura baked hardtack before going, and they ate that with most meals. Neighbors gave them (or very occasionally sold) them food as well, and they were sometimes invited to stay for supper. Almanzo and Mr. Cooley also fished when they had the opportunity. Laura also records when they passed wild fruits and berries.
• Farmland: Almanzo and Laura were farmers, and they had a good eye for the farm and pasture land through which they were passing. Laura faithfully recorded the condition and nature of the crops they saw, in addition to comments from the locals on the weather and price of land. Commentary on land price and quality is the single most consistent theme in her account -- not too surprising, really, when you consider that they were making this big journey for the purpose of buying farmland in the Ozarks.
• Horse condition and behavior: The horses' health was an overarching concern, as was the possibility of their being stolen; upon learning of a band of horse traders up ahead, they made sure they weren't going to camp anywhere near them.

A few representative quotes:

• "Mrs. Cooley and I went to a house to buy milk. It was swarming with children and pigs; they looked a good deal alike."
• "An emigrant team is behind us and every minute I expect to hear the usual, Where did you come from? Where are you going? How are the crops up your way? This never -- hardly ever -- fails."
• "The streets (in Topeka) are asphaltum pavement and it is lovely to drive on, so soft and quiet that it doesn't seem real. It gives like rubber to the horses' feet. The caulks on their shoes make dents in it and slowly the dents fill up till the place is smooth again."
• "Started at 7:20. Found a little black-and-tan dog in the road, lost. He is skin and bones, must have been starving, and is afraid of us. We stopped at several houses to ask, but nobody knew where he belonged so we are taking him along. The children delight to feed him milk. We have named him Fido."
• "Water has been scarce all day and what little we found tasted so bad we could not get a good drink. It is clear and clean but it feels slick and tastes bitter, it spoils the taste of tea. The horses have to be very thirsty to take it."
• "Got a good start at 7:30 but the roads are awfully stony. Crops are poor. Everyone tells us they never get rain here when they need it ... at noon we came to Mound City which is quite a city. We bought bread and an 8-cent pie and 2 cents worth of tomatoes. Tomatoes are 30 cents a bushel."

Rose Wilder Lane (Laura's daughter) wrote a foreword and afterword to the collection, and I looked through a lot of her references (like Coxey's above) and was surprised to discover how accurate she was. Then again, a lot of people studying the Wilder books have wondered if it was Rose, and not her mother, who was the real genius behind the stories, but that's a discussion for a different day.

Edit: Fixed spelling/grammar errors and clarified the point re: Laura's commentary on land prices.

For a weeks-long journey by a wealthy traveler, would they bring much of their food with them, or was it common to just pay locals to feed them along the way?

For your time period, basically the same things people do on long road trips today: talk to others in the carriage, read, try to sleep. Get out periodically to stretch, look out the window.

Obviously, long expeditions across more "wild" country took more extensive preparation, but for, eg, traveling across 18th century France it was not particularly dramatic, although admittedly with a higher chance of having to deal with bandits than the typical road-trip across France in the 21st century.

While I'm not sure what you mean by "elaborate dress" there were such things as traveling clothes.

Read Horace Walpole's memoirs. He explains how he cried for weeks after one of his puddles was eaten by a wolf when crossing the Alps in a coach, and how he shot several thieves in Cornwall and had the rest hung after they indicated the wrong road though a forest and ran after him to attack him during the night. So basically when not speaking about almost anything, travelers would maintain their guns, play with pets, play together (Piquet ?), read, write long letters, draw impressions of what they had seen (no cameras then) or improve sketches taken on the field, etc. They could have very long exchanges about genealogy, ancient history, theology, the Bible, and all kinds of science, and exchange gossip about nobility and infinite money problems.

In 'An Innocent Abroad', Mark Twain's weeks on the ship seem to consist largely of activities to keep the travellers distracted: dinner parties, games like dominoes, diarising, mock courts, and things of that kind. There was a lot of socialising, as you'd expect, and the travellers were very familiar with one another. This helped Twain's comedy through the book.

They frequently stopped at ports along the way and went out during those time, such as at the Portuguese Azores.

Just be bored.

In Mark Twain's All Along the Equator[?] he writes about riding a stagecoach across Nevada and only mentions boredom, heat, and dust.

Medieval historian here. There's a pretty interesting document called The Disputation of Mallorca that depicts a Jewish and Christian merchant having an ongoing discussion/debate about their two faiths over the course of their travels and commerce together. It's still debated whether this particular document is an actual recording of a real conversation that took place, but the point nevertheless remains that people would probably just talk and hang out as they traveled about their business.