shotguns...oh how I love them !
Ok...I'll try to answer some of your questions here. The "choke" of a shotgun barrel is the pinch or restriction at the muzzle( the end). It's small, just hundreths of an inch but directs the shot (pellets) coming out of the gun into a tighter pattern. Some guns have removeable chokes that unscrew and are called choke tubes. A "full" choke will give a tight pattern and keep the pellets close in a 25 yard shot. If you wanna see it, blast away at a piece of cardboard at 25 yards. When you put in other chokes like modified or cylinder, they basically restrict the barrel less and let the shot spread into a larger pattern.
As far as barrel length...18" is good for home defense, might work for skeet, and is basically useless for trap. Skeet is less formal, you go to a field with your box of clay birds and a thrower and blat away and have some fun. Trap is regulated and the station you shoot from is I think 16 yards from the trap box and the bird is fired from a good mechanical thrower which is usually faster than throwing by hand with skeet. I shot trap competitively for years and the serious shooters were using 28" barrels and even a few 30", most with full chokes for a nice tight pattern.
$200 for a shotgun ain't much. I'd recommend buying a decent used gun rather than a new cheapy. My first choice is always the Remington 870 pump. Second choice would be either a Mossberg 500or Winchester and lastly a Savage or Ithaca. Those would all be the pump shotguns. There are automatics but for a first gun, a pump is a nice start. Simple, reliable, easy to maintain. Remington makes an 870 Express that you could probably find used easily and new in places like Walmart for not much more than $200.
Pawn shops are another good place to find a decent gun. I saw a practically brand new 870 by me for $195. Never been fired, had a scratch on the stock.
Regarding slugs....it's been a long time since I've fired slugs so I'm gonna take a shot at this and may be corrected by others. Since most of the guns I've had were full choke, I was told shooting a regular slug down it could damage the choke. The alternative was to use a sabot slug. Basically, a sabot slug is slightly narrower with a paper or plastic sleave around it that won't hurt the barrel. Using the Rem 870 as an example again...IF you wanted to shoot slugs, for instance, for deer hunting...you can very easily by unscrewing one retaining nut/cap, can remove your 28" vent rib trap barrel, and install an 18" or 20" rifled slug barrel with either rifle sights or scope mount. If it takes you more than 30 seconds to change a Remington 870 barrel, you're either asleep or dead.
Good luck
. It might be wise to use a longer barrel though. 
I actually always wanted an 870 ever since I knew what it was. Thanks again guys.