Taipei is a stunning and vibrant city, characterized by friendly people and safe streets. The downtown is almost divided into two areas -- the west side, with its narrow streets and road side vendors, and east Taipei with mega malls, chic boutiques, and classy restaurants.
It's very easy to find a Taipei hostel -- there is an excellent array of cheap accommodations scattered around the city. When looking for a place to stay, usually the question is "where" -- what's important is to find something near the MRT or in an area that has the kind of restaurants and shops you like.
Leaving your Taipei hostel and moving around is easy -- the MRT connects almost every place in and around the town; it's one of the best in Asia and it's cheap, safe, and very clean. One-day passes are a good deal; they allow for unlimited rides on the metro and associate bus services. There are also many buses, but they are a little difficult to use, because destinations are mostly written in Chinese.
The city is great for eating out -- leave your Taipei hostel and visit the most common places for outdoor dining at Taipei's night markets (Shilin is the most famous). Here you can try every kind of finger food at street stalls, or if you prefer, you can have a seat in one of the traditional restaurants. Food is good, varied, and cheap.
The city offers so many attractions that you'll probably need more than a couple of days' stay at a Taipei hostel to visit everything. Historic sites, old streets, temples, hot springs, shopping malls, and new skyscrapers -- Taipei has something for everyone!