Treat your new tattoo like a fresh, open wound. Throughout the healing process, your tattoo will go through different phases. In the first few hours, the tattoo will ooze plasma with blood, I recommend washing it with water and antibacterial soap regularly the first day.
Do not neglect your tattoo. Make sure you clean it with water and antibacterial soap after any physical activity like playing sports, running, cleaning your house, etc.
Let it air dry for about 20 minutes before applying any wound ointment. Make sure your hands are clean and apply a thin layer of ointment 2-3 times a day until it heals.
Within a week or two, your tattoo will peel and feel itchy. Do not scratch or pick your tattoo. Instead, apply some moisturizer and gently massage to ease the itching sensation.
Do not expose your tattoo directly to the sun. Do not submerge your tattoo in the pool, beach, bath/jacuzzi, or any dirty water. Remember, it is an open wound and can get infected.
The better you take care of your new tattoo, the better it will look when it is completely healed.