Chainring and cassette compatible?
Assuming Shimano components.... You shouldn't have any difficulty with compatibility.  However, as noted above, you need to find out if you have a cassette/freehub or a freewheel.
At 20 years old I imagine you're looking at a freehub...
And likely 7 speeds.   If you want to go to a more-modern 8-speed cassette, the spacing should be the same but you'd need new shifters.
The crank may or may not have replaceable chainrings, and if it does they may not be easy to find.  Nashbar used to sell a wide variety of chainring sizes and bolt-patterns, but current catalogues do not show these.
Likely available from outlets like Loose Screws.
Note that some cranks may appear to have replaceable chainrings but they are in fact permanently attached and you have to replace the whole crank.