For a pump action you can't go wrong with a Remington 870, a Benelli Nova or Super Nova would also be a good choice. For a semi auto I don't know but I'd start with a Remington 1170, also look at Winchester, Browning, Beretta, etc.
I don't know what to suggest for a barrel length, you're going to have to decide what will work for you. The shorter barrel "Home Defense" shotguns don't typically have a interchangeable choke system (they have a "cylinder choke" and you would be unable to replace it with a tighter choke which you would want for skeet shooting), a typical hunting shotgun with a 24" barrel is going to be awful long to try and maneuver inside your home. Personally I'd go with an 18" or 20 " barrel, it will be fun to plink around with and would be a good HD gun, then get a nice over/under for shooting skeet.
In regards to the comments above about recoil between pump and semi auto: Yes, on paper there is a fixed amount of recoil for a particular round no matter what you're shooting it out of, in practice however, a semi-auto has much less perceived recoil than a pump action because the recoil is spread out over a longer period of time and used to operate the action instead of all being absorbed by your shoulder all at once. I can load up my Winchester Super-X 2 with 8 full power slugs and put them on target as fast as I can pull the trigger. The same rounds out of a pump shotgun are brutal, even out of a Benelli M4 (also a semi-auto but inertia operated rather than gas) they are bad. A pump action is simple and may be more reliable than a semi-auto but it's still not fool proof, under stress especially, you can "short stroke" the pump and still have problems.
