By 1987, Schwinn had switched over from all the oddball proprietary sizes on its parts, so you can use pretty much anything you want. Just make sure the bike is a good fit before committing a lot of time and money to it. That goes for components, as well. I hate it when I have collected a bunch of parts to build a bike, only to find out that something doesn't work right with the chosen combination.
I understand wanting to do a project just for the experience and satisfaction. I did that with my Fuji S12-S LTD. I didn't start out with painting in mind, but when I got the bike home, I saw that it had been rattle canned and had homemade decals, so I decided to jump in with both feet. I stripped it down to bare metal, repainted in my preferred color, applied new decals, and built it back up. I kept as many original parts as I could, and similar parts for the rest. I get a good feeling whenever I ride it because of the work I put in to make it my own.
1981 Fuji S12-S LTD