What kind of lenses can I get for my Nikon D5100?

I'm starting to expand my hobby and am purchasing the Nikon D5100. Assuming I decide to purchase the kit lens (18-55mm) I would like to add one more. For price reasons I'm looking at Tamron instead of Nikkor.

Ideally, I would like to get something with the largest range possible, I just don't know exactly what I need to look for to be sure it is compatible (I know there's something about the autofocus motor or something?).

If anyone can suggest a specific lens or a few that would be wonderful, don't have to be Tamron (but the cheaper the better!)

Added (1). I'm not a beginner with DSLR cameras… I have used my dad's Nikon for a few years now and have just decided to purchase my own. I know that I want an additional lens, 55-300 or similar, and am not just purchasing one because I've been told to do so but because I have experience with the basic 18-55.

Stick with the 18-55mm for now - a DSLR and standard lens is a big enough handful for someone just starting out.

Once you get used to what you're trying to do, THEN look at supplementary lenses.

If the Nikon costs $500, and the Tamron costs $350, you are wasting $350 to save $150. Doesn't make a lot of sense to me.

Always buy Nikon lenses (or perhaps Tokina - and even in a pinch - Sigma, but never Tamron). Even if you have to save up a bit longer for them, the lens will not only be better, resale will be better to, so you are protecting your investment.

You will obtain better performance if you buy the less expensive Nikon D3100 and use the difference for a better lens. A good lens is more important than a good camera.

A good camera + cheap lens = cheap photo.