I'd recomend a pellet gun over a BB gun. BB guns are generally made with plastic, have crudddy trigger pulls and poor sights. They also are generally smooth bore, because bb's are steel, and not exceptionally accurate.
A pellet gun on the other hand, should have a rifled barrel and will shoot lead pellets which are shaped to be much more accurate. They also have the benefit of not ricocheting quite as easily (they still will though, so be careful what you use as a backstop).
You don't have to spend a ton of money, but I would get as nice of a gun as you can. Gamo, and Benjamin Sheridan both make very high quality guns for the money. They aren't top of the line, but they are very good bargains. I'd look at a gamo CFX, or 220 as a very good gun to start with. They have great triggers, adjustable sights, and the 220 I have right now is very accurate. They might be a little bit too powerful or too loud though depending on where you live. Gamo makes other guns that are less powerful if thats the case.
Sheridan makes a really nice pump up gun. They don't offer as much power as the spring piston guns from gamo, but have variable power (just very the # of times you pump the gun). The sights are a little bit lacking, and the triggers aren't always great. But its a very solid, reliable gun that will last forever, and has the potential to be very accurate with some practice.
If you think you'd like to spend a little more, look for a Beeman or RWS gun.
I'd also recommend you think about buying from
www.pyramidair.com (handle the gun at a store before hand if you can). They have the best prices I've found online so far, also the widest selection of guns and ammo. Sporting goods stores have good deals once in awhile, but don't always have what you want, and don't necessarily have the best price, you have to do some looking around.
I don't want to come across like I look down on anything that shoots BB guns either. I have a couple and they can be fun to get out once in awhile. But for someone that wants to learn shooting, I think a higher quality gun is much more rewarding, and a much better purchase in the long run. You don't want to limit your potential as a shooter by having a gun thats less accurate than you are.