Many basic plumbing problems, such as a running toilet or dripping sink, can be repaired by the homeowner.
The best way to decide if you can tackle the job is to use Google to search for the specific problem you have. You will find Google will produce countless articles and directions on how to fix most minor plumbing issues. If any of them seem out of your league then you can always contact a licensed professional as a backup plan.
It may be a good idea to have a few tools on hand in case you need to do some emergency plumbing work. You may think you will never do this, but remember that plumbers are not always going to be able to come the exact moment that a situation arises. You should at least have a pair of pliers (preferably plumbing pliers) and a few different sized wrenches. These can be used to tighten and adjust leaky plumbing for a temporary fix.
At a minimum, make sure you know how to turn off the water supply to each of your fixtures and to your main house. That way if you have an emergency leak you can shut off the water supply until a plumber can come and save the day. You will find the shut off valve for the toilet and sinks coming directly out of the wall under these fixtures. Simply turn this valve to the "off" position to stop water from coming to each fixture.
The main shutoff valve to your house is most likely outside by your front curb. Often times it is in a visible metal box in the ground. You will need a pair of pliers to switch this valve to the off opposition. In some cases you may be able to shut off the main valve to your house in your basement or crawl space. If you are confused as to which valve is the main water supply you should make sure that you have someone show you where your home water shutoff valve is located. This knowlege can save you thousands of dollars in water damage repair should you have a water leak.