A shotgun is good for home defense, but not so good for "target shooting". You can shoot skeet or trap (those little flying clay frisbees) with a shotgun though, which is tons of fun. If you want to use the same shotgun for both Home Defense and trap/skeet, you probably want to buy an extra barrel: one short barrel for home use, one long barrel for sports. Most brands have extra barrels available, and some even sell both barrels with the gun as a package deal.
If you want a rifle that can be shot in an indoor pistol range, and is also good for home defense, the Phenom is definitely right. Check your NYC/NYS laws to see what the complications are, but I believe that most models are legal so long as they don't have a folding stock, flash suppressor, etc. NY residents please correct if I'm wrong, and I'll edit my post.
Popular rifles in pistol calibers (which are inexpensive to shoot, can be shot in indoor ranges, and may be less problematic for self-defense shooting in an apartment) include rifles such as: Beretta Storm, Ruger PC, Kel-Tec Sub2000, etc. You can run a "Search" on this THR forum to find other writeups about these exact models. Also, the AR-15 (civilian version of the M16) is available in pistol calibers, but I believe that you have to buy a post-ban version to be legal in NY.
If you want something cool that looks a little more mainstream than the "Evil Black Rifles", then definitely check out a lever-action rifle, like John Wayne used to carry. Holds plenty of ammo, comes in nice short versions for easy handling, affordable and extremely durable, and might be less off-putting to friends that you take out shooting. Just as lethal as the above semi-autos, just different aesthetics. Good lever makers include Marlin and Rossi/Puma/EMF, with the latter being sold under several different brand names. The R/P/E/etc. are also lighter and smaller than the Marlin, being very good copies of the 1892 Winchester.
http://www.emf-company.com/1892-winchester-rifle-carbine.htm
Just a few ideas, hopefully some NYC folks will weigh in soon. In future topics, it may help to post that you're NYC in the Title.
Welcome aboard,
-MV