First and foremost, do NOT listen to your friends or internet "experts" in regards to healing your tattoo. This is the aftercare I was taught, and it's been working fine for 20 years.
Please leave your bandage on for one hour (minimum), but not overnight. If the bandage feels a little stuck when you want to remove it, hop in the shower and get the bandage wet before removing it so that it won't rip/tear the skin.
Wash your hands thoroughly with VERY PLAIN FRAGRANCE-FREE soap (Ivory is as plain as it gets) and warm water. Wash your tattoo, using your hands only, not a washcloth nor loofah. When the skin looks smooth and free of dried blood and any leftover goop, let it air dry for a few minutes, and then apply a very small amount of ointment.
Do not overdo it with the ointment, you want your tattoo to be moisturized, not smothered. If it looks shiny, sticky, juicy, or wet, you have too much product on there, so wipe it off. Too much product can cause more problems than it can help, so be aware.
Wash your tattoo once daily, and use ointment 2-3 times a day, for 2-3 days. After 2-3 days, STOP using the ointment and switch to a very plain, fragrance-free lotion (Lubriderm and Eucerine both make fragrance-free products.) Keep your body wash, shampoo, conditioner, and perfume away from your new tattoo.
DO NOT pick or scratch at your tattoo. It will peel, it may get a little scabby, it is going to itch. You can either slap it and/or put some lotion on it. No picking!
NO SUN, whether it's real or fake bake, for a solid month. AFTER one month, please apply spf 30+ anytime the tattoo will be seeing the sun. The sun is the enemy of tattoos; protect yours against the damaging UVA/UVB rays and it will look good for years to come.