When it comes to getting things done in any number of situations, most people don't just go with whoever is available. You won't ask just anyone to help fix your refrigerator, and the same goes for your tattoos. Because tattoo artists are just that: artists, and because all artists have their own sense of style, not to mention strengths and weaknesses, it is within your best interest to do some shopping around before just getting a tattoo.
A good thing to start with is to have a solid idea of the kind of tattoo design you want done. Then, check out what sorts of tattoo parlors are around. Ask friends with tattoos -- hell, ask strangers who've gotten tattoos locally. Check out sites online that rate various tattoo parlors in your area. The first step is to be at least moderately informed and to then be prepared to do a little searching.
Take the time to visit the local tattoo shops and talk to the artists there. Ask to see their portfolios, and find out if they are more comfortable taking a pre-made design and then tattooing it, or if they prefer to design a tattoo by hand based on a customer's specifications.
If you've already got the design on hand, the choice of tattoo artist isn't as important, as long as the person can replicate existing material well. In the case of asking for a specially-made design, don't be afraid to ask for a portfolio -- a true artist will be glad to show off their handiwork! Pay attention to the sorts of designs are inside the portfolios you ask about, and make sure what you're looking at goes well with what you are looking for. Because if you're looking to get a tattoo of Death riding a motorcycle made out of bones, it might not be a good idea to go with the artist who does a lot of unicorns and fairies, and vice versa.
It is also important to find not only a tattoo artist you are comfortable with, but to find a tattoo parlor you feel comfortable in as well. It is a good idea to ask about sterilization and, if it's your first tattoo, to ask questions about what exactly to expect. Because at the base of it, each of these parlors is ultimately a business, anywhere that you may not feel comfortable asking questions may not be worth handing over your money, even if you like what you see.
All in all, finding the right tattoo artist is almost as important as finding the right tattoo to begin with. Because your tattoos will often last longer than you will, so you might as well make sure you're going to be happy with it. Not just for a week, not just for a handful of years, but for the rest of your life.