The tattoo bandage keeps air-born bacteria from invading your wound. Open flesh is a breeding ground for bacteria and infection. You should leave the bandage on for a minimum of two hours.
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. 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.
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. 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!
After your tattoo is healed, from now on, you will always want to protect it from the sun's ultraviolet rays. These can fade and damage a brilliant tattoo very fast. Before spending a lot of time in excessive heat, protect your tattoo with a minimum 30SPF sunblock.