Swapping 1994-95 Diamond back sport response components out
It looks like it runs a 3x7 Shimano mountain drivetrain. You can get new shifters such as the Shimano ST-EF51's (the 3x7 version; which are switchable between V-brakes and cantilevers -- see manual for details) or the SL-M310 (in the 7 speed version for the rear, triple for the front; add your own brake levers if yours are integrated with the appropriate cable pull). I'd probably opt for the SL-M310's personally -- I've had the ST-EF51's briefly and the quality left something to be desired.

Alternatively, you can buy the Microshift DS85-7 (grip shifts) or TS38-7 (trigger shifters) and add your own brakes. Or SRAM Attack! grip shifts. 

If you're really cheap, just buy a set of thumb friction shifters. Any ones will do. These aren't indexed so you have to move them into the right position to shift based on feel.

I'd also suggest replacing the cables while you're at it.