Your tattoo artist applied a bandage to 'cover up' your tattoo for a good reason. To keep nasty bacteria entering your wound. The skin has been opened therefore it is considered a wound. Whenever flesh is opened it is a breeding ground for all types of bacteria which cause infections. The bandage should be left on for at least 2-3 hours. Please try to resist taking it off, I no you are eager to look at and show people your brand new tattoo, but it is vital that you keep it covered up.
After the 2 or 3 hours is "up remove" your bandage and wash your tattoo. Make sure to use warm water with a mild 'antibacterial soap'. Gently wash away any blood or ointment completely cleans the entire area. Just use your hands and not a cloth, cloth could contain bacteria, just make sure your hands are squeaky clean and your good to go.
After the area is completely clean use a 'clean towel' or "paper towel" and pat the area dry, do not rub, just pat.
If you would like to purchase 'product Tattoo' Goo or H2Ocean is recommended. Though not necessary, many over the counter products will work just fine and just as effective, but it is up to you.
Continue to keep it clean, use lotion if you prefer or ointments, to keep the skin nice and soft. Just remember make sure the "lotion" is dye and fragrance free, many 'tattoo parlors' recommend Lubriderm, some people have reported it stings, so beware of that. I myself use Eucerin.
It is perfectly fine to take a shower with your "new tattoo". Its ok to get your tattoo wet, but don't soak it. Other words don't take a bath for at least 2-3weeks, soaking your tattoo can cause damage, so just take a shower and your good to go. If soap or "shampoo" gets on your tattoo just remove it as quickly as you can with some water. Also don't swim for 2 weeks.
After a few days you may notice some scabs or peeling, a lot of scabbing could be a sign of poor tattoo work, but a little scabbing is sometimes normal so don't panic. The best way to rid these scabs is to apply warm compresses to the scabs for at least 6 - 7 minutes 3 times a day this will soften the scabs and they will peel off on there own. Do not use a lotion or ointment to soften them. More then likely they will itch please try not to pick or scratch them, if it gets real bad just slap it hard and that should minimize the itching.
The suns ultraviolet rays can fade and even damage your tattoo quite rapidly. Just remember if your going to spend a lot of time in the sun, weather it be at the beach or a long walk make sure to apply a sun block to protect your tattoo against the suns ultraviolet power.