Let's assume, for simplicity's sake, that we're talking about a knight in Northern France (which, for better or worse, is usually the archetype for European society in the central middle ages). The important caveat is, of course, that things varied a lot depending on geography and time period - knights in Scotland, Poland and Sicily would have had very different experiences, despite an emerging common chivalric culture.

So, assuming you're a wealthy knight in one of the great northern French principalities, like Anjou, Champagne or Flanders, you would probably own quite a bit of your own land and would live in a manor house or small castle on those lands. (Poorer knights, and often younger knights, would often form part of a more powerful lords's household and live with him for most of the time.) You would have have inherited this land, probably from your father but equally possibly from a maternal uncle, grandfather or more distant relative. The 'absoluteness' of your ownership would vary depending on the time period: around 1000, it would be your 'allod', your family land, but from c. 1100 it's more likely that one of your lords, a more powerful man to whom you've pledged fidelity, would 'allow' you to inherit or would formally confirm your inheritance. This was due to an increased blurring of the lines between 'allodial' lands and a type of land know as a benefice, which was held either temporarily or permanently from a lord, whether lay or ecclesiastical.

All of this land was the source of your wealth, and so much of your time would be concerned with managing it. An essential part of this management was maintaining relationships with your neighbours, your lords and any dependents you have yourself. Holding a benefice wasn't the simple 'land for loyalty' contract that it's often portrayed as (except in England, where things are weird). They were relationships with obligations and expectations on both sides, and were constantly being negotiated. It wasn't simply a two way relationship either; there were frequently contested claims and there were often rival lords who would be willing to support rivals. Deaths and inheritances were usually the spark for conflicts and feuds over land and rights could last for years. A famous example of a relationship gone sour is a dispute between Count William of Poitiers and Hugh of Lusignan in the early 11th century, which frequently spilled into violence and involved frequent accusations of betrayal and deception on both sides.

A great deal of trust was involved on all sides in these relationships, therefore, which was forged through shared participation in the local political community. These sorts of events, such as attending lordly, ecclesiastical, comital, ducal or royal courts with your peers are a key way to influence decisions, build up relationships and win influence. Providing counsel to those higher up the nobility was a vital duty at these assemblies and, while not quite a vote, a lord was honour-bound to accept and act on the counsel of his faithful men. Assemblies and courts were also the location for lots of legal decisions and transactions, and acting as a witness or a go-between in these transactions (overwhelmingly about land) was another way to build relationships and trust.

Much of your time would also be spent in church, in prayer or around clerics. Although there was obviously variety in levels of individual piety, the church was a big powerful player in the medieval world and, as such you would spend a lot of time engaging with the church in the sorts of legal and political gatherings discussed above. Many of these big assemblies took place on major church feasts, in the local cathedral or major monastery, and you would be expected to attend to be part of the action. You would likely also have your own religious devotions too, either in a private church or chapel (what historians call eigenkirchen) or, from the twelfth century on as private churches were taken over my monasteries or converted to parishes, a personal chaplain. You are also quite likely to have family members in the church, which was a common career choice for younger sons. You might at some stage of your life undertake a pilgrimage (or, after 1096, a crusade), particularly if your lord were doing the same.

All this political management of your social position, what historian Stephen D White calls 'feudal politics', would have taken up much of your time, but there was more mundane management too. Rents and dues from your land needed collection, minor disputes resolving, the household needed managing and your patrimony needed consolidation and expansion (which, contrary to popular images of conquering your neighbours land, was usually done through sales, leases and mortgages). Much of this work might be done by your own dependents, but if you were on the smaller scale of landowner, you might need to do it yourself and you might even need to manage a part if your lords' lands on his behalf (although you'd get a cut of the revenues in return). All of this wpuld have involved quite a bit of paperwork, which would have been written up by a cleric; writing was a specialized skill, although you could well be able to read basic Latin as a layman. We, sadly, have no surviving non-royal lay archives from before the very end of the 12th century, but we know they existed and it's using this family archive that you would manage your lands and finances.

You might be thinking now that this politicking administrator sounds nothing like the knights we think of in the middle ages, but warfare was still an important part of the knight's world. It's unlikely that you would need to perform guard duty or patrols (they're for less important people) but you would be expected to keep in fighting shape. This was primarily done through your leisure activities however, which were very martial: horseriding; hunting; and from the twelfth century, training for and participating in tournaments. These leisure activities were conspicuously expensive and exclusive to the aristocracy and a way of displaying your status as well as staying in shape for war. You might often do them with your lord or peers, further strengthening those feudal relationships. Entertaining and feasting was also an act of conspicuous consumption, especially at major holidays; at such feasts you'd hear poems songs and stories from the emerging chivalric culture; Roland, Lancelot and Ruodlieb would all reaffirm the aristocratic values of loyalty, piety and aristocratic privilege that defined your daily life.

If you are a wealthy noble knight, it's because you have been granted an estate by your lord. Managing this land would be your main task, and military service its price. As a minor noble, you have to deal with finances, farming concerns, management of the peasants, maintenance of your manor house, participating in several levels of politics, maintaining or expanding your holdings, properly settling your children, visiting neighbors and family, etc. And since violence is a major part of your way of life, you'll also be managing your arms and armor and horses, training yourself and your horses and pages and squires, quarreling with your neighbors or whomever offended your honor recently, traveling to and participating in tournaments, and participating in various duties to your lord (including his squabbles and obligations).

However, there were times and places where you can find knights (or men who looked like a knight) in circumstances rather like you describe, with the general exception of "wealthy". In the early middle ages, these would be household men of a lord whom he would arm and use for whatever violence he had in mind, such as riding about the countryside terrorizing the peasants or quarreling with a neighbor or serving his lord's various needs for violence. As the cost of equipping heavy cavalry became increasingly expensive, it became easier to give your servants land and make them responsible for the cost, or force your existing minor vassals into armed service.

Wealthier lords would continue to employ a retinue of armed men. Some of these armed servants (and "knight" derives from a germanic word meaning "servant") and family members would become "knights" as well, to be equal in status to the minor landed noble class of vassal knights as described above. Household knights for wealthier lords (like kings) might occasionally be well rewarded, and might sometimes have managerial duties, but were generally professional warriors. Later in the middle ages knighthood would became a noble title somewhat divorced from expectations of military service (as it is completely today), and you would see men-at-arms who would perform the military functions of a knight (heavy cavalry, mostly) without the status. These might be household men, or professionals, or yeoman, or mercenaries.

Of course, this all varies by time and place, and can get messy. The ideal feudal model hardly ever applies perfectly.

For more details, I might suggest:

The Knight and Chivalry (1996). Barber, Richard.

Chivalry and Violence (1999). Kaeuper, Richard.

Medieval Knighthood V (1994). ed. Stephen Church, Ruth Harvey.

Although this was not universally true, many knights elected to participate in tournament. Tournaments started in roughly the early 12th century and evolved greatly over the remainder of the Middle Ages. In the 12th and 13th centuries, tournaments primarily took the form of melees, wherein two or more teams would get together in a field and duke it out. The mechanics and ceremonial aspect of tournaments are both very complicated and David Crouch does a far better job in Tournament than I could of explaining them but in essence these early tournaments started with individual jousts called commencailes early in the morning (which eventually evolved into tournaments made up entirely of individual jousts) and then later, usually in the afternoon or rarely the next day, to the melee. In the melee, a knight and his squires would try to capture other knights and ransom their persons and take their horses (which could often be returned on the payment of further ransom, see The Life of William Marshal translated by Nigel Bryant and Crouch's book on tournament). For a wealthy knight, this loss of money or equipment would be easier to stomach than for his poorer companions. For example, Edward I racked up a debt of 70l. (pounds) worth of horses when tourneying in France in the 1260s (Crouch, Tournament, 106). Tournaments were immensely popular among knights both because of their potential as sources of income, and for their pageantry.

In the thirteenth century particularly in England, tournaments could involve bits of Arthurian pageantry with various magnates taking on the roles of different Arthurian heroes such as Gawain, Lancelot, and Kay (Edward I deployed these as a sort of propaganda for his imperial plan in Wales and Scotland and often cast himself as King Arthur).

Sources:

David Crouch, Tournament.

Juliet R. Barker, The Tournament in England 1100-1400.

Nigel Bryant (translated), The Life of William Marshal.

As a corollary/possible clarification because I'd love to see this answered, what would be similar/different for this moderately wealthy knight in High Medieval England, France, Germany and Poland (just for fun)?