First of all, take good care of your tattoo and of your skin in general.
Your tattoo artist will give you instructions on caring for your tattoo. Be sure to follow them, because he or she will know the best procedure to follow.
Keep the bandage on as long as the artist tells you to. Wash the tattoo area with antibacterial soap only after you have already cleaned your hands very well, also with antibacterial soap. Then apply an aftercare product, as advised by the tattoo artist. (Keep in mind, however, that if you have any allergies, you should avoid using products that contain those problem substances. You will want to discuss this with your tattooist.)
Re-apply the ointment several times each day, but don't overdo it. You can actually cause fading by applying too much and too often, so discuss the correct application with your tattoo artist as well. Wash the area several times a day also. Once the healing process is finished (usually two weeks or less), you will want to use a moisturizing lotion to keep the area soft and keep the colors looking alive.
While the tattoo heals, don't pick or scrape at any scabs that may form, or scratch your skin when it itches (tattoos can cause quite a bit of itching - don't give in to the urge!). Color and ink can come off with any scabs or flakes of skin you pull off, so you can really damage your tattoo by scratching at it. It is also best to wear loose clothing to avoid irritating the skin and the loss of ink.
Avoid strong sunlight - cover your tattoo if you are going to be out in the sun a lot - and stay away from swimming for the first two weeks as well. The chlorine used to keep pools and spas germ-free can bleach out your ink, and rivers or lakes can harbor germs that may cause an infection in the tender skin of a new tattoo.
For more details on caring for your new tattoo, see our FAQ on "Taking Care of Your New Tattoo."