Actually, most integra parts will swap over to a civic, but I haven't heard of many people going the other way. For simplicity, let's just say the integra got better factory considerations.

Trouble spots to look for [rust and such] are under the rear bumper, in the floor areas and fender lips.

You can be pretty sure that any car you find in good condition is going to have some very soft suspension bushings though, so between bushings and tires, I would say those would be your first considerations for competition prep.

Another good thing is that all the integras are fuel injected, if you don't like to tune much. Only the Si's were fuel injected in the civic family. The integra had DOHC non-VTEC 1.6's, and the civics all had 1.5's of some configuration [CVCC carb'd, or non-CVCC MPFI]. There's plenty of hop-ups for either though, just remember that the civics and CRXs are lighter also.