Clean the tattoo, and apply any protective ointments you are using to help it heal correctly.
Put on a dark and loose shirt that you are positive is clean. Dirty fabric touching a fresh tattoo can cause it to get infected. Be sure the fabric is dark, so that any ink that transfers to the fabric will not stain it, and loose, so the tattoo can continue to breathe even though it is covered.
Peel the fabric from your skin anytime you feel it stick to the ointments or lotion covering the tattoo. This will help to not suffocate the healing skin.
Remove the shirt as soon as possible, and allow your fresh tattoo to heal, unfettered.
Put on a dark shirt with sleeves long enough to cover the tattoo completely, and keep it covered even when your arm is in motion. A shirt may be necessary for larger tattoos, but smaller ones can be covered in other ways.
Place a tennis sweat band or ace bandage over the tattoo if it is on your wrist or forearm and small enough for the band or bandage to hide completely. An ace bandage can certainly cover more surface area, but you will most likely have to field several inquiries. You can either make up a phony injury or tell the truth; either way people should be pretty interested.
Place a small adhesive bandage over the tattoo if it is extremely small. Most people don't even notice adhesive bandages anymore, as they are pretty commonplace.