What lenses would be best for this situation?

I have the nikond5100 with the VR AF-S 18-55 kit lens.
I think I want a telephoto.
I was wondering if a brand other than nikkor fits nikon cameras.
If I get one without the VR would that be ok as I mainly wish to shoot some horse shows (quick moving and jumping horses) and perhaps some nature/architecture shots from traveling?
I don't have a very high budget so what lenses would fit these types of shots better than the kit lens without breaking the banks? (The cheaper the better although I don't want something so cheap that its not worth having, if you know what I mean.)

Added (1). Oh, those would both be in my price range.

Why do you want to buy a generic lens?

The two possible lenses you can buy to extend your zoom range would be the Nikkor 55-200 mm VR ($200) or 55-300 mm ($400)

Blue lenses is better

If not Nikon then Tamron, Tokina, or Sigma.