Removing a tattoo, regardless of whether it's new or old, large or small, is expensive and time-consuming. Laser treatments by a dermatologist, surgical removal or dermabraision are the only safe, proven methods of getting rid of them. You may be able to reduce their appearance somewhat by exposing them to sunlight and swimming, or by using an over the counter product called Wrecking Balm.
There is no foolproof method of removing a tattoo, even if it is freshly inked, without medical help. However, when you get a tattoo done by a professional tattoo artist, there is a list of things the artist will advise you against in order to protect your new investment. If you choose not to follow the tattooist's advice, the likelihood of the ink fading somewhat is good.
The first thing that a tattoo artist will advise you against is getting the tattoo wet. Once the tattoo has scabbed over, then it's there forever unless you seek medical help. However, if it has been less than a week since you were inked and you are beginning to regret your decision, you might want to go swimming. Chlorinated water on a fresh tattoo might cause it t fade.
The sun is another tool that might help fade the tattoo. Artists will advise against exposing new tattoos to the sun. If you are trying to get rid of your new tattoo and you don't swim, see if sun exposure will fade it a bit.
Another good way to fade a tattoo might be to expose it to salt water. A trip to the beach will expose your new ink to all the things you've been advised against: Water and sun.
If you have tried all of the above and none of them have worked, then you might want to try a commercial product such a Wrecking Balm. Again, there's no guarantee that it will work, but it's cheaper than laser treatments. According to the manufacturer, your success rate with Wrecking balm is going to be determined by the size of the tattoo and how many layers of skin it has penetrated. The fewer layers of skin, the more likely you are to be successful at removing it, the package says.
There is no guarantee that any of these methods will work for removal of a new tattoo. If you want foolproof results, see a dermatologist for laser removal.