I catch them all the time - we are overrun with them and possums. Same method works for both...
I use a live trap of a size and design similar to this:
http://www.kness.com/Shopping/shopexd.asp?id=28
Although mine is a cheaper version of it. For raccoon you need one about that size though. it will work for opossum as well.
Every time I see one around, I set the trap out, and rarely come up empty.
A few tips... I always leave the trap outside, and it gets dirty and rusty, but it doesn't get human smells on it.
Every time I use it, I lube the moving parts with vegetable oil. and set it for a "hair trigger"
For bait I use either an ear of feed corn, or a partly peeled sweet potato or regular potato - use a zip tie to tie it to the back of the trap pan, where they HAVE to step on the pan to get at it. Once it's tied in, smear it all over with peanut butter. Tie it in the center of the pan toward the rear so that a coon can't reach in from the sides and get a handful of peanut butter - that's what they want. I've never really tried other baits, because peanut butter smeared on something works every time.
Try to set your trap in such a way that there are obstacles on the sides - to discourage the critter from coming up on the sides of it and trying to reach in. You want the easiest route to the bait to be through the front door. A narrow ditch is a great place, because it limits access from the sides and funnels them through the door.
Some traps like this have a release door on the back (opposite where the critter comes in). If you buy one, I would look for that feature, because it makes it much easier to bait. Mine doesn't have it, and it's a PITA to tie the bait in reaching in through the front door.
When you catch one, he won't be very happy or friendly, but if you take him somewhere far away and turn him loose, he won't come after you or anything. You may have to shoosh him out of the cage, but once they realize they are free, they haul butt away from you as fast as possible.
I don't really relocate mine... I have two cur dogs that I hunt with - they tree squirrel and coon, so when I catch a coon, I just take him a hundred yards out in the woods and turn him loose and set the dogs after him. After he's been through that he'll be more wary of my place. Coons that live in town can get pretty bold, but my two curs keep 'em honest around my neighborhood.