Being your own plumber is fraught with dangers. You could easily find yourself on the wrong side of the law, risking a fine and breaching the terms of your insurance, and if the job goes wrong, it could cost a lot to put it right. Although the law varies from state to state, it is generally illegal to carry out alterations to the drinking water supply, the sewage system or anything involving gas, unless you are a licensed plumber.
There are, however, a few jobs that are suitable for DIY. Here are a few bathroom repairs that you should be able to carry out yourself.
Unclogging a bathtub drain
Bathtub drains are not usually subject to the grease that can be washed from a kitchen sink, but they can still get clogged with hairs and other debris. You will need to use a drain snake to do the job properly; simply take off the drain plate and use this to access the drain with the snake.
Install a new shower head
This is a very simple task -- the old head should simply unscrew from the shower, and the new one can be screwed on. It is a good idea to clean the threads on the shower arm before screwing on the new head, and wrap some Teflon tape around them (although you should check the instructions on the new showerhead, as some do not need the tape).
Fix a leaking tap
Taps may simply need new rubber washers or may need the whole mechanism taken out. Although there are several different types of tap that you may find in your bathroom, they should simple and safe to repair. Just remember to turn off the water supply to the tap -- there should be valves near the taps themselves; otherwise, you will have to turn off the mains supply.
Fix a running toilet
Problems with the valve in the toilet can cause it to run continuously, rather than simply refill the tank and stop. The valve may need to be adjusted or replaced, but this is a simple job which you should be able to do yourself.
For any more complicated bathroom repairs, however, your health and safety may be at risk if you do not use a licensed a plumber. In that case, you should call in the bathroom plumbing professionals, and get the job done properly, safely and legally.