The trick to caulking tubs and showers is using the right type of caulk. There are many different kinds of caulk out there that are used for home repair, from filling cracks in drywall to sealing off plumbing appliances. Make sure you check the label, and only get caulk that is designed for tubs and showers, or ask someone at the hardware store for help. Go for quality when you purchase a caulking gun. They aren't that expensive, and they are easier to use and last longer.
A leaky faucet can be a relatively easy fix depending on the type of faucet, so get to know the basic types of faucets and how to take them apart. You can look for instructions online; YouTube videos are the most useful. A slow drip usually means that all you have to replace is a washer, but make sure you know which size you need before buying a new one.