It really doesn't make any difference how the connection occurs between the tub and the tub and the shower walls as long as the joint is properly caulked. Having said that, the big deal is getting rid of the old mildewed caulk and properly applying a new layer. Use caulk solvent to get the old stuff out. Then clean the connection with a scrub brush and lots of chlorine bleach. Rinse with water and use a hair dryer to completely dry everything out. Wait 24 hours and apply a new coat of silicone caulk to the joint. Wait the 24 hours even if it means renting a hotel room to take a shower. Caulk will not bond to a wet surface or where water vapors are present.

The next time you clean your shower walls -- make sure that they are perfectly spotless -- apply a coat of car wax. The wax reduces surface tension and makes cleaning easier. Some folks use pure lemon oil instead of the wax, but we like the wax.

Since you just moved in you should call a plumber to deal with your clogged drains. Let him or her show you how to access all drains, show you where clean-outs are and help you establish a maintenance procedure. You'll only have to pay the plumber once. Learning to do things right the first time will make you feel good about establishing (and following) a sound maintenance routine.