Good food on the cheap in Taipei?

You can get cheap and good food at Gongguan station which is near National Taiwan University, lots of great and cheap food for college student. 

However, if you have more dinning budget (about $7~$14 per people）, I'll recommend restaurants near National Dr. Sun Yat-Sen Memorial Hall and Zhongxiao dunhua station, which is so-called "Eastern District ", one of the most cosmopolitan parts of Taipei. 

Food is generally inexpensive in Taipei. 
There are many many inexpensive and delicious places all around - the trick is getting to them!

I strongly suggest getting unlimited 3G data plans at Taiwan Taoyuan International Airport (TPE), Chunghwa telecomm is a great option.
Google maps will make you feel invincible in Taiwan.
Likewise you can walk, take the train, or cycle (free for the first 30mins) to your destinations.
Cab is fairly cheap as well.

The first that comes to mind are the night markets:
Shilin Night Market (the biggest most expansive)
Shida Night Market (the cheapest , being located near a university)
Raohe St. Night Market (regarded the best generally)

However, if you get sick of night market food, (not that difficult really)
These are my usual go to options:
Yongkang St (many many awesome food choices here! Try Yongkang Beef Noodles, the original Ding Tai Fung, James Kitchen, and many more..)
Gongguan St (Student friendly places with tons of hip and inexpensive eateries)
Good Cho's (Another interesting places to for fancy bagels) 

Good idea to check out local english blogs for ideas as well.
Cheers.

You can find great food all over Taipei, especially if you venture off the main streets just a little.

One of our favorite beef noodle shops is just off the Zhongxiao-Fuxing intersection on the north-east side, on Andong St. 

For night markets, big fan of Raohe St and Gongguan Night Market, where you'll find the best brown sugar bubble milk tea Chen San Ding, which is right next to a great Gua Bao (pulled pork belly sandwich) shop.

The are many early morning breakfast stalls on the southern end of Fuxing Road, neare the Technology building station.

In general, I'd suggest wandering, looking for small restaurants with a good size crowd.  Often times, either there will be a menu with pictures you can point from, or sometimes there's an English translation.  Our go-to dumpling and beef noodle shop just happened to be just that type of find a block or two down from our apartment on Jinhu Rd. in Neihu. 

Be sure to look for a case somewhere with the appetizers they may have, and pick up some of the slightly pickled, sometimes spicy cucumbers, and the boiled peanuts.

If you are not going to restaurant, foods are cheaper. You can also go night market/

Shilin Night Market will be a great choice for you!