If it's a brand new tattoo it should be covered up for a couple hours and than let to heal out in the open air. You want to keep it out of direct sunlight and don't let it soak in water.
Keep it clean by using scent-free soap and letting water run over it. Pat it dry, don't rub it.
In a few days it will start to get itchy and may flake or peel. All of that is completely normal and a part of the healing process.
It's important to remember that a tattoo is a lot of scratches in your skin that take longer to heal because they have had ink inserted into them to creat the picture. Treat them as such, keep them clean and let them heal.
You can wear a t-shirt that covers it up but there may be some bleeding or excess ink that will stain the shirt. It may also irritate the skin if fabric is rubbing agasint it.