After getting a job, your priority should be finding a place to live. Luckily, Taiwan's cost of living is tremendously low, and the city is very convenient to live in.
Typically, foreign English teachers just move into some apartment that's been rotating foreign English teachers for many years. If you want to start from scratch in your own private apartment or deal with Taiwanese realtors, this post isn't for you.
Getting an apartment
Your best, and possibly only, tool will be facebook. Search for "roomates in Taipei" or "apartments in Taipei" to find the facebook groups that foreigners use to trade their rooms when they're moving out.Typically, posts will have the price, some pictures of the apartment, distance to the nearest MRT, and other features. You contact the person through facebook or phone, and schedule a time to visit.
The most typical living situation is in an apartment with one flat rental fee that is split based on room size between the roomates. Generally, you will deposit the rent monthly (for the next month) at a bank.
When I was in Taipei, rent was anything from, per person, 7000NTD - 14000NTD per month.
Whether or not you deal with the actual owner of the property is a toss-up. It was 4 months before I was ever put on the lease of my apartment, and that's just because the lease needed to be renewed anyway. Many times, the foreigners just pass the deposit along the line, so you will pay your deposit directly to whoever's room you're moving into. Make sure to get a receipt.
Before deciding on a place, get all of the details. How do you pay rent? Will you be put on the lease right away? How is communication with the landlord? Check to see how far the MRT is, consider your travel route to work every day, if you know where you'll be working already. Ask why the previous person is moving out. Living situations in Taiwan can be very fluid, but it's still important to be guarded.
If this post seems scarce, that's because it really is that easy. I wracked my brain trying to think of anything else you need to know, but honestly, this is it.