Two somewhat simple things come to mind. 1) Most likely, your shower arm is leaking behind the wall. Unscrew it (yes, it will unscrew out of the wall), then put some teflon tape on the threads and screw it back in.

2) Your tub spout is also attached poorly. It may also unscrew, or there is a retaining nut under the spout that has to be loosened so that it will slip off the pipe. Either way, remove it, pack the hole in the wall behind the spout with plumber's putty, use teflon tape on the threads, then screw it back on. If it's a slip-fit type, you might want to replace the spout with another slip-fit type since the o-ring that keeps it watertight may be damaged.

I should have mentioned to be careful not to scratch the finish on the pipe. One way to unscrew the shower arm without damage is to first remove the showerhead, then use the handle side of your pliers and stick one handle into the pipe to give you more leverage. Brace the pipe with your other hand as you try to loosen it.