Properly caring for your tattoo is important to its look. You're just as responsible for your tattoo looking great as your artist, so be sure you take the right precautions to be sure it lasts and looks great for years to come!
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 awesome 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 one hour.
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 as soon as possible. You're better off not having any covering than to be suffocating your new tattoo with plastic wrap for a long period of time. It was only placed there to keep it protected until you are ready to care for it.
After you remove the bandage, you will want to wash your tattoo. Use lukewarm water and mild, liquid antibacterial or antimicrobial soap.
Dial tends to be too harsh and abrasive - 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 while washing - when the plasma dries on the skin surface, it creates scabs.)
Then pat (do not rub) the area firmly with a clean paper towel to get it completely dry. (Do not use any other sort of towel) Follow with a very light application of your choice of ointment. AQUAPHOR ointment or AQUATAT would be the ideal first choice, but if you don't have any, Bacitracin or a similar antibacterial ointment is acceptable.
Wash and apply ointment every 6-8 hours for the first few days.
It is important that you use very little ointment!
**Do not use Neosporin! This is a great product for cuts and scrapes, but not for tattoos. Some people can even have an allergic reaction to it, which causes little red bumps. When the bumps go away, so does the ink, and you end up with a polka-dotted tattoo. There's plenty of other options you can use that will work much better and are less problematic. **
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 and continue for 3-5 days.
After it starts to scab, continue to keep it clean, but use plain, non-scented lotion when needed instead of ointment, to keep the skin soft. Whatever lotion you use, it should be completely dye and fragrance free. Your tattoo should fully heal in 3-6 weeks.
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 and use only your hands when cleaning it. Swimming - whether it be a pool, fresh water or salt water - should be avoided for at least 2 weeks.
If for some reason you absolutely need to submerge the tattooed area for some other reason, coat it with saran wrap and thoroughly tape the edges to keep the area free from water. Try to avoid this if possible. When out in the sun, be sure to use sunblock and do not go tanning.