pass the ASVAB and based on your score you get to select through certain jobs. You don't have to focus on the mechanical part of the asvab unless you want a job as a mechanic.
I applied through a recruiter. The process took 4 weeks. I interviewed at US Army (Dallas, TX) in Aug 2014
Interview
go to your nearest recruiting station and the recruiters will handle the rest. it's not rocket science. if you have what it takes to serve your country and don't mind them taking all the time they can from you then this is definitely the career for you.
ROTC/Service Academy/OCS are the only ways you can become and officer in the united states army. If you enlist, you have the option to become an officer through a variety of different ways like Green to Gold or OCS training later on after basic training.