Caring for a tattoo is actually really easy and simple, there's really only two important steps and a bunch of do not(s).
STEp ONe: CLEANSE
It's extremely important to make sure your new tattoo stays clean while it's still in the healing process to avoid any potential infection. It's highly recommended by many tattoo artists to use unscented soap to clean your tattoo as fragrances can lead to severe irritation and potentially cause scarring and even ink loss due to the chemicals used in fragrance soap.
I personally like to use the Dove Unscented Soap because it has moisturizing ingredients in it to help prevent dryness of the skin, which is a bonus if you're using it on a new tattoo. I wash my new tattoos at least twice a day, every day, until I'm sure they're completely healed (about three weeks).
Also, don't forget to make sure your hands are washed before touching your tat. This also means don't let your friends, family, or coworkers touch your new and healing tattoo.
STEP TWO: MOISTURIZE
After cleansing my tattoo I always apply lotion to it, also make sure the lotion is unscented too. I like to use Lubriderm on my tats because I find it to be soooo moisturizing and it lasts long so I don't have to apply it as often. I also find that it helps calm itchiness (because god damn do healing tattoos itch!!!)
Don't use Vaseline. Although many tattoo artists use Vaseline while working on tattoos, it's really not that good of a product to use for after care. Vaseline is 100% petrolium which means it doesn't absorb well at all. If you noticed, your artist put Vaseline on your new tat before bandaging it so that the bandage wouldn't stick to your new tat.
You want to make sure you use something moisturizing, that absorbs quickly into the skin and allows your tattoo to breathe to speed up the healing process.
If you have any questions regarding tattoo healing, leave a comment below and I'll be sure to get back to you!