To make sure your tattoo is clean, and to help speed-up the healing process, you'll need to make sure you take the time to gently wash it and moisturize.
To do this, rinse the tattooed area softly with lukewarm water, and a non-fragranced pharmacy soap (don't use regular bar soap or shower gel, it can cause really bad skin irritation around a fresh tattoo).
Never rub the tattoo with the soap and be sure to really slowly, and softly, pat-dry the tattoo with a clean towel or even medical gauze.
Be sure that the tattooed area is dry before adding any ointment or lotions to the area. This will help it get absorbed into the skin and help your tattoo heal faster. Don't put too much ointment or lotion on, it's actually better to use less and apply it more often.
You'll need to repeat this process at least once (maybe twice) a day for about the first two weeks.