Actually, when I did that conversion I used a Shimano Freewheel. Since your bike is an '89 it might have a cassette. Here is a good Sheldon Brown description of the two types:

https://www.sheldonbrown.com/k7.html

The article also describes the removal installation process. If you don't have the tool, most bike shops will remove cassettes/freewheels for a couple bucks. I've even had them removed for free.
I like the SRAM cassettes, but shimano are good too. That really is all there is too it. Since you're replacing the rear cassette, get a new chain too. The Park Tool website has good videos on how to adjust the rear derailleur for the increased rear capacity.

https://www.parktool.com/