Sorry to disappoint you if you thought you had a classic, but your Vivitar is probably not very old at all. In fact it is still listed as a current product!
It is however an "old-style" manual focus 35mm SLR so I can see why it would seem old.
Aaaanyway...
The lens mount is the "K mount"... this is Pentax's lens mount, however it has also been used on cameras by third parties including Ricoh, Cosina, Chinon and Vivitar.
You could find an inexpensive second-hand telephoto prime in shops or auction sites by looking for ones by those companies for the K-mount.
Alternatively if you want new lenses, look for Pentax, Sigma, Tamron or Tokina (again for Pentax K mount).