It would depend on where you are geographically, but assuming "somewhere in Europe" as a decent constant, the first thing you'd notice would be the much smaller armies. (Byzantium does things very differently mostly because they were able to keep up their land tax system--u/hesh582 has an excellent answer right below mine as an informative contrast.)

These are generally made up of noblemen and their retinues, each of whom is more or less "self-funded" out of their estates instead of being paid from the central state fiscus like you are. This in turn limits the ability of the king to compel them to do things like "pay taxes" or "serve for 20 years", so there are few if any of the big, disciplined, combined-arms, professional-infantry-centered, expensive full-time armies going around. Medieval battles tend to involve hundreds to thousands; the Roman battles often saw twenty thousand or more combatants to a side. Without the kind of centralized state apparatus that the Romans had, the typical Medieval ruler simply can't afford the legions, and instead has to rely on warlords who swore their allegiance to him (and smaller warlords who swore allegiance to them, et cetera).

The smaller retinues also tended to be mounted. I am not sure of the exact balance of infantry to cavalry (and this varied from place to place, naturally enough), but the trend is pretty clear that while the professional legion infantry were the heart of the Roman armies, mounted knights were the central part of most Medieval armies. In individual training, these knights are probably no worse than you and your legion buddies; in most cases the knights would have plenty of time to train themselves while their peasants and tenants did the actual farming. What they won't typically have is the kind of disciplined cohesion drummed into the legionnaires. Legions had a corporate existence and infantrymen who were loyal to their legion and to their buddies (the modern term is "primary group cohesion"); medieval knights usually did not serve in that kind of long-service organization, instead swearing loyalty to one or another master, who was usually their landlord.

The shift away from large numbers of professional infantry to smaller numbers of cavalry means that you are no longer given pride of place on the battlefield. Infantry were usually held in somewhat lower regard, so you might be ordered to hold a shieldwall or spear schiltron, but the battle is usually expected to be decided by the crash of armored cavalry. Open battle, though, is not all that common. 1000AD sees the rise of castles, privately owned forts, and sieges of those castles will happen a lot. You as a legionary will probably be familiar with the basics, but what might shock you is the decreased sophistication of siegecraft. Rare is the early Medieval army that can pull off the encirclement-and-circumvallation that Caesar used to smother Alesia, and the massive siege ramp that was used to take Masada is more or less out of the question. Siege engines never died out completely; the catapults and ballistae throwing rocks and projectiles are going to be generally familiar enough, and the wooden siege towers, scaling ladders, and battering rams will all be things that you know how to work, but the scale of the sieges are going to be junior-league stuff.

Also, a word about logistics. As a Roman soldier, you would've been used to having one of the most sophisticated bureaucratic organizations in Europe up to that point backing you up to make sure that you could be deployed where the Republic/Triumvirate/Empire/Pompey [delete whichever is inappropriate] needed you. This included a standardized organization where every eight men had a mule and (often) a camp servant or slave, an allotment of rations, a fairly standard set of equipment, and all that was needed to keep it running. This "table of organization" goes up through centuries and cohorts all the way to the legion itself. Then there were the incredibly huge baggage trains, often several trains deployed in parallel, keeping the legion supplied while it marched. Now, this did break down on campaign from time to time, but the impressive thing is how often it worked. As a foot soldier, much of this would've been above your pay grade, though, and it would probably come to your notice when it breaks down and you're forced to go hungry--which seems to have happened much less with Roman legions than it did with other armies. The sheer amount of money that Rome could throw into its wars paid dividends.

All of that is gone in the Medieval army. Instead of the centralized professionalism and amount of money that Rome could field, logistics becomes a haphazard affair, and whether or not you'd go hungry depends largely on how successful your individual warlord is at organizing things. If you're used to Roman organization, this would probably come as a rude shock. Goodbye to standardization. And to add insult to injury, you probably wouldn't even be *paid--*unless you were part of the mercenary corps floating around, hired from campaign to campaign and generally despised by the mounted nobility, you were generally expected to make your living as a tenant or vassal of some nobleman somewhere, being provided with land and serfs in exchange for your service.

A workable parallel is this: imagine you were a member of a modern, professional military. Your paycheck comes (mostly) on time, there are regulations and procedures, there are clear lines between what belongs to the army and what belongs to you, and if everything is bogged down in bureaucratic miniutiae, at least things get done in a routine, standard way. And then, suddenly, you find yourself working with some random warlord. The weapons will be familiar; the tactics need adjusting for the much smaller unit sizes; some of your new teammates are good shots and probably excellent fighters. But everything that you rely on the army to provide for you--food, ammo, shelter, transportation--is now being handled by some relatively wealthy rando. If you're very lucky, he can keep you properly supported and supplied. But chances are that he can't, and you are going to have to adapt to very different, usually lower expectations.

(You'll no doubt have noticed that I didn't go so much into personal equipment. Most of it hasn't changed so much as to be unrecognizable in the intervening thousand years. The byrnie or mail hauberk that the knights wear will be familiar to you as a longer lorica hamata; the swords and spears might incorporate relatively minor design changes, and the shields tend to be round or oval instead of rectangular, but it'll be nothing that you can't understand.)