To get an apprenticeship you need to become a regular at a local tattoo shop. It will require a portfolio of your artwork and finding someone willing to apprentice you. There are no schools to go to, it's all on the job training.
Apprenticeships, like school, are not free. Prepare to pay a few thousand dollars for the privilege of learning how to tattoo. The duration of an apprenticeship will vary depending on who you get to teach you and how well you pick up the art of tattooing. You might be able to find someone to teach you the ropes in 6 months, but if you expect to have a real professional teacher who will expect you to be able to tattoo proficiently you're probably looking at more like 1-3 years.
Also you really want someone who has alot of experience. in cases that ive found, most wont apprentice unless they feel as though they have enough exprierence. Because of the fact that you will be carrying on thier name. also you have to know the artist, in my personal opinion i wouldnt apprentice anyone that just walked in and asked for it. they have to trust in you and your ability for the art. but Admin, did pretty much put the nail in. i just wanted to seal up the coffin.