Basically, what a compound bow does is it uses pulleys to release the tension off the string after you've pulled it past a certain point. The let-off can range from 50% to as much as 90%, but the average is within the 65% to 80% range. So say if you have around 75% let-off on a 65 lbs bow, then you're only holding about 16-17 lbs of pull once you pass the initial draw. This will allow you to hold a heavy-pull bow steady for aiming or stalking, for much longer periods of time.
The feel of a compound bow is very different though. You don't have the steady draw weight. Instead of a more fluid motion from pull to release, a compound bow (which will usually be a much heavier pull), will require you to throw your entire body into pulling the string at first then once you get past the let-off you can then take your time to aim and shoot. The recurve therefore makes a better quick-response type bow that you just pull and release in one fluid motion. The compound is a bow which you prepare slightly ahead of time before your shot.
If you're going to be using a compound in the field, then you should practice with a compound. If you're going to use a recurve in the field, then you should practice with a recurve. They are quite different shooting styles.
So basically think of it this way - would you rather nock your arrow with your bow strung but not pulled, then when you see your prey you raise, pull back, and release... or would you rather stalk your prey under cover, then when you see him you prepare your bow, get him in your sights, and wait for the opportune time to release? Or... would you rather be a gunman or a sniper?
It all depends on your personal style.
If you want to take down big game, then that's usually where people will want a compound over a recurve. The let-off system allows them to carry a much heavier pull than they would normally be able to in a traditional style bow. Recurves tend to be better for small game, giving you quicker response. Since you're into this for the archery and not the hunting, then I would surmise that this is probably a moot point for you. Thus, I would lean towards the recurve as probably being a more comfortable choice. Especially if you're a martial arts student, you are probably more interested in the traditional methods of draw.
I don't think I would suggest a straight bow (aka "longbow") unless you want to re-enact a certain period with it. A recurve, especially a laminated one, simply allows you a stronger pull by virtue of better design. Shooting style is the same, but the straight bow is just weaker (of course I mean for a bow of similar size, weight, dimensions, etc.).
