I'm not up on the nightlife or hotels. Google searches ought to be useful for both. There's no nightlife district as such, thought. There are clubs scattered about. The "eastern district" is pretty lively but its mostly restaurants and shopping.
The airport-downtown trip is a bit of a schlep. There are various kinds of buses for NT$100-140, which is reasonable. One leaves for the Taipei RR station, where you can switch to city buses and 2 lines of the subway. Several others make 3-4 stops in different parts of the city. Once you know you hotel, you can decide on your bus (the hotel can probably tell you). Once you are in the city, taxis are a reasonable way to get around to many places, but you need to speak Chinese or have the hotel write down where you are going.
Of the touristy things to do, the National Palace Museum is required viewing. Try to go as early as possible and on a weekday. Temples are fun to visit--both local temples and the big ones like Longsi (Lung-ssu), Xinggong (Hsing-kung); also Baoan gong (Pao-an kung) is next to the Confucius temple. Some other museums are interesting, like the contemporary art museum. All these things are in the regular guidebooks.
Also, you need to visit a night market or 2. Your hotel can tell you nearest one. In general, eat at a variety of small restaurants (they'll be cheap); night markets are for eating too. ....And as in any city, just walk around (Taipei is a relatively safe city though of course the usual cautions apply). If you are a night person, there are a number of 24 hour restaurants (Fuxing South Road just south of the Howard Plaza Hotel is famous for them); also 24h bookstores like the Eslite (Chengpin) on Dunhua South Rd. Good for people-watching.