Well, if you go the handgun route I suggest either a .22lr or a 9mm. A .22 if you want the handgun purely for target shooting and for the reason that it's good to start small and cheap. If you don't see your self purchasing another handgun in the future (yeah right) and you may want to use the handgun for home/self defense hence the forum you're in, I would suggest a 9mm to start. Cheap(er), soft recoiling, and great ammo availability.
If you go .22, I would suggest a Ruger mkII/markIII, or mark 22/45. These are all essentially the same gun in one variation or another. They are well made, accurate, and relatively inexpensive. If you choose the 9mm, I would suggest either the S&W M&P9 or the Glock 19. These guns are also relatively inexpensive, although more than the .22's but, they are accurate, reliable, and most important for a new shooter is simple to use. There is no safety to fiddle with (read up, they are safe or, you can get the S&W with a manual safety if you choose), there is no decocker or single action/double action to learn. You simply load the weapon and shoot. Great handguns for target or home/self defense for around $500+/- used.