Worn-out washers are the main cause of these leaks and they are inexpensive to replace. Knowing the brand name of the faucet that you are replacing parts and washers from will help insure a proper fit.
Washers are pretty simple to install, but when in doubt call a plumber.
If a shower head is leaking make sure it is screwed tightly, if that doesn't work unscrew, clean both the threads on the shower arm and showerhead. If you have some Teflon tape wrap a little around the shower arm. Now screw on the showerhead and make sure it is tight. Also, remember to check washer for wear. If you need a new washer you can get one at you local hardware store or from your plumber.