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.
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.
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.
Scabbing and Peeling
After a few days, you will notice some peeling and possibly a little scabbing. Excessive scabbing could indicate a poorly-done tattoo, but a little is sometimes normal and there is no need to panic. Apply warm moist compresses to the scabs for about 5 minutes 2-3 times a day to soften them and they will eventually come off on their own. (Do not apply ointment or lotion to a softened scab - wait for it to dry) You will also start to itch, just like a sunburn when it begins to heal. The advice here is, don't pick, and don't scratch! If the skin itches, slap it. If it is peeling, put lotion on it. And if it is scabbing, just leave it alone. Your tattoo is almost healed, and now is not the time to ruin it!