Okay, I'll explain how it works step by step.
1) You'd have to check the exam listing. One test occurs about every 5 years. The link is below:
http://www.nyc.gov/html/dcas/html/employment/examschedule.shtml
2) Apply for the exam with the instructions they give you. You will have to pay $40 or so to take that exam.
3) You will receive an admission card in the mail a few days before the exam date.
4) Go take the exam.
5) Within a month to a year, you will receive your list number and exam score. You'll have to wait to be called. You will have an interview, a medical exam/screening, and something else I keep forgetting...
Side Note: Having connections with an MTA employee will not get you in the MTA whatsoever.