There are two main ways I like to heal a tattoo: second skin and dry healing. I prefer to use the dry healing method for bigger pieces. But I do use the second skin for the smaller pieces.

For dry healing: You should keep the clingfilm barrier on your new tattoo for at least two hours. You can keep it on for longer if necessary, but not for longer than 12 hours.
Once you unwrap your tattoo, wash it with your clean hand using warm water and a bar of gentle, antibacterial soap. I recommend either Protex Gentle, or Dove.
Let it air-dry for about ten minutes or so, for most pieces. For larger full colour pieces, you can either wrap it in clingwrap again for another day, or let it dry out completely for two days before using any kind of cream. I will advise you on the best way to go.
Once dry, smooth on a *thin layer* of the recommended aftercare. It is very important that you use a very thin layer, as you do not want to smother the tattoo. You use this aftercare twice a day for two weeks. You can use it for longer if you like, it only makes your skin healthier.

It is important to note you cannot go swimming, or bathing with a new tattoo. You should take quick showers, so that the tattoo stays clean, but does not spend lots of time in water.
DO NOT SCRATCH. It will get itchy as the new skin grows in. Do not scratch it, or it will heal badly. Leave it alone.

In the long term, sunscreen is now your best friend. Keep it out of the sun as much as possible, and look after your skin. Remember that a tattoo is part of your skin, and should be looked after as such.