Lynn, sorry but I thought you were talking about exterior moldings. With interior moldings, the biggest part of the decision depends on how you want it to look. If the paint that's there is not too heavy, wash with TSP (available from hardware and paint stores, then sand mainly to rough up the surface, then repaint. If there are obviously 900 coats of paint on it already or you know that there are hardwood moldings that you want to expose, then strip. To strip, use a product called Peel Away, available from places like Woodcrafters. To use this, you glop it on then cover with their special "paper". Wait 24 to 48 hours, peel the "paper" and the stripper and old paint will come off. Double wrap in old newspaper and discard. A few localities have some special rules on discarding so I can't help you there but the old paint probably has lead in it. You might have to scrape off some of the remainder also with a putty knife or small tools but most of it should come off. Then wash with TSP, lightly sand and repaint. It is OK to paint over walls made of drywall--which they didn't have 75 years ago--or just about any other kind if inside walls. You might want to spackle (interior) to fill any holes, wash the walls with TSP, prime with something like Kilz then repaint. The problem with painting is that it is 75% or more getting ready to paint but the 75% makes all the difference in the job. If you use good paint, it will last a lot longer and I prefer buying at the specialty paint stores. The people there can help you with any special problems you might encounter or let us know if you have further questions.