My recommendation would be for an over/under or a semi-auto in a 12 gauge. Find a place to go shoot and see if they rent guns. If they do, try some different guns out and see what you like. They will likely rent over/unders, but ask and see if they have any semi-automatics.
Browning and Beretta are among the more popular "middle of the road" shotguns out there. Their semi-autos range from $600 to $2000 (or more depending on additional options like a recoil reducer). Their over/unders start at $1500 and go up from there. If you are looking to get into trap, skeet, and sporting clays "casually", then I would recommend a Beretta A391 Urika 2. You can get the sythetic stock version of this gun for $900. This is a very solid gas operated semi-automatic shotgun and will give you many years of trouble free service provided that you clean it and take care of it. I've heard of people putting 1000 rounds a day through this gun without any mis-feeds or jams.
If you find that you like the over/unders but don't want to shell out the money for a Beretta, Browning (or Winchester or one of the other more expensive brands), then take a look at Yildiz. They are a Turkish manufacturer and I've heard some good things, plus they are very budget friendly.
If you are truly interested in something that will do double duty as a home defense gun and a trap/skeet gun, then I would pick up a Remington 870. You can get both a standard length barrel as well as a short, 18in barrel for home defense. Load it with 00 buck for home defense. The problem I have with this is that if you find you really get into trap/skeet, you'll grow out of that gun rather quickly.
The last thing I would recommend is picking up a shell bag to wear on your belt (some have belts included). Much, much easier than carrying around boxes of shells or stuffing them in your pocket.
Good luck!