There are a couple of different ways to achieve this goal. There are those that join the military and get their training compliments of Uncle Sam. You can go to an FAA approved school and get your training there. Another way, would be to find an internship or apprenticeship program with a Repair Station and get hands on experience.
To get an FAA Airframe and Powerplant (A&P) license, you must have at least 30 months of training or hands on experience in order to be allowed to take the test. That is a general, an airframe and powerplant test, followed by an oral and practical test. Once you pass those test and have your FAA license in hand, you will be in very high demand. If you have specialized knowledge (like avionics and electronics, non-destructive testing, composite materials, heavy structure repair and manufacturing) you might be able to get a Repairman Certificate from an FAA Certificated Repair Station, to perform those special tasks. The difference is that an A&P license goes with you wherever you go, while a Repairman certificate stays with that Station.
As Mr. Ursani mentioned, this is not always an easy 9 to 5 job. There is a lot of responsibility, and the conditions are not always ideal. However, there is something infinitely rewarding about watching an aircraft you worked on all night take off into the sunrise. Among Aircraft Maintenance Technicians (AMT) we always joke that you can have the best pilot in the world, but without a good technician, that aircraft will never leave the ground. Every person onboard that plane is depending on your craftsmanship, quality and dedication to safety.
I have always loved airplanes and I have always been very mechanically inclined. My parents did not always approve of me taking my toys apart, but that's how my training started. Joined a model airplane club and built lots of models. Got my FAA A&P license compliments of New York City Aviation High School. Went on to College of Aeronautics (currently Vaughn School of Aeronautics) and got my Bachelors degree. The big airliners never really attracted me, so I stuck with General Aviation. Loved helicopters and the private and corporate jets.
Join Professional Aircraft Maintenance Association (PAMA) and get a subscription to AIN (Aircraft International News) magazine, they both provide a wealth of knowledge. Research technical schools near you and see what training is available. Stay true to your dreams and follow thru. Good luck!