Lodging: Zhongzheng District (AirBnB)
A hidden, tucked away AirBnB with kitchen, bathroom, and bed and a street view window cost $57/night, but don't forget about the $16 cleaning fee and $17 service fee + $6.50 in taxes (all USD). Still, with the comfort and seclusion of this place, you will not be disappointed. Right out of the lobby door you enter the street markets. Breakfast/Lunch/Dinner is only a few steps away!
Taipei is a fairly big city and the best way to navigate it is to have your EasyCard loaded and ready for using the bus or subway. Unfortunately, renting share-bikes is not possible for foreigners or even Mainland Chinese! So, good walking shoes are a must and cash for food to fuel up your body is essential. If you're from the US and have a SunTrust credit card, then there is no worry about foreign transaction fees (but watch out for ATM withdraw fees adding up).
For those who enjoy being outside and hiking, it is highly recommend you do the small hike known as "Tiger Mountain" (as opposed to the potentially more crowded Elephant Mountain). These two hikes are nearby and somewhat hidden, but Google Directions will work just fine. Along the way and on the way home, there are so many foods to try in the streets of Taipei. For another day of walking, I definitely recommend taking a stroll along the Tamsui River to take a break from the tight and sometimes crowded streets.