What does it take to go from E-2 to E-3?

Do your job and wait. Your Team Leader will push for your waiver if you deserve it.

Your commander is the answer and/or 1SG recommendation and approval to the commander. Usually time in grade is 12 months TIS (time in service) and 6 TIG (time in grade) to make PFC. There were lean times back in the day where no one was waived (promoted earlier then those timelines) due to budget and cuts to the Army. Some things you can do to increase your chances at receiving a waiver: be in the right place, at the right time, in the right uniform, study and attend Soldier of the Month boards, be motivated and rocking your MOS, proper military courtesy and respect at all times. If you do these simple everyday Soldier things with a good demeanor and outstanding first impression, you can make SPC before 26 moths.

special note: the only two ranks in the Army up to YOU, SGT and SSG! E-2 thru E-4 are up to you company chain of command, all others above SSG are DA select.

Good luck.

You've already been given some solid information from previous commenters so I will add only what hasn't been stated (unless I missed something). Once you arrive at your unit be sure to surround yourself with the right people who are doing the right thing. The military gets all types just like any other workplace. You want to be around those that care about their career, have the ability and desire to help you learn (and vice versa as time goes on), and are highly regarded by their peers and their leaders. If you do all the things mentioned by everyone here you will get the results you are looking for and more. It might be sooner, it might be later. The point is you are setting the standard everyone will come to expect from you throughout your career in the military.