Leave That Bandage Alone!
Your artist took the care to cover up your new tattoo for a very good reason - to keep air-born bacteria from invading your wound. Yes, as pretty as your new tattoo is, it is still a wound. Open flesh is a breeding ground for bacteria and infection. Leave the bandage on for a minimum of two hours. Excitement of having a new tattoo will make you want to remove the bandage so you can show your friends, but your friends will just have to wait until later.
The only exception to this rule is if your artist covered your tattoo with saran wrap or some kind of plastic. This is extremely detrimental to a tattoo, so it should be removed immediately. You're better off not having any covering than to be suffocating your new tattoo with plastic wrap.
Wash and Treat
After you remove the bandage, you will want to wash your tattoo. Use lukewarm water and mild, liquid antibacterial or antimicrobial soap (Satin and Provon are my highest recommendations. Dial tends to be too harsh - generic brand antibacterial soaps are actually better) to gently wash away any ointment, blood and/or plasma and to completely clean the area. Do not use a washcloth or anything abrasive. Your hand is your best tool in this case. (If your tattoo feels slimy and slippery, you have probably been oozing plasma. Try to gently remove as much of this as possible - when the plasma dries on the skin surface, it creates scabs.)
Then pat (do not rub) the area firmly with a CLEAN towel or paper towel to get it completely dry. Follow with a very light application of your choice of ointment. A&D vitamin enriched ointment would be my first choice, but if you don't have any, Bacitracin or a similar antibacterial ointment is acceptable.
**Do not use Neosporin. This is a wonderful product for cuts and scrapes, but not for tattoos. Some can have an allergic reaction to the Neosporin, which causes little red bumps. When the bumps go away, so does the ink, and you end up with a polka-dotted tattoo.**
Specialty Products and Lotions
If you prefer, you can also use a specialty product such as Tattoo Goo or H2Ocean. It's not necessary, as many over the counter products work just fine, but it's your choice. Use the products as directed as continue for 3-5 days.
After that, continue to keep it clean, but you can use lotion when needed instead of ointment, to keep the skin soft. Whatever lotion you use, it should be dye and fragrance free. A lot of artists recommend Lubriderm, but I have found that Lubriderm stings when I apply it. Instead, I have had great success with Eucerin. See also: Aftercare Contradictions: Why there are so many views on what product is best.
Bathing, Showering, Hot Tubs, and Swimming
Yes, you can (and should!) shower with a new tattoo. It's OK to get your tattoo wet - just don't soak it. Submerging your tattoo in a bath or hot tub can cause serious damage, so you'll want to avoid those for 2-3 weeks, but showering is perfectly fine as long as you don't saturate your tattoo. If you get soap or shampoo on your tattoo, just remove it quickly with water. Swimming - whether it be a pool, fresh water or salt water - should be avoided for at least 2 weeks.