Definitely visit Taipei 101. There's some cool shopping there and it IS the tallest building in the world currently, so it's something to say you've done.

As far as getting to downtown from TPE, a cab will run you about US $20 from TPE - Taipei 101. I've done this several times. I've also taken the bus, which is significantly cheaper (~ US $5). And the buses are fairly nice, sort of like a domestic F seat, rather than the city buses that you are used to in the USA. Ask at the information desk at Arrivals, they will direct you down a hall to where the buses leave. You can see which buses leave to what locations. There are several bus lines that will stop in front of most of the main business hotels and main tourist sites. You can look on the sign on the wall (in English) and see which bus goes where as well as prices. It's very easy. I'd recommend taking the bus if you want to save $15, especially if you are alone and you don't have too much baggage. If you are with 2 or 3 people, you might as well just take a cab, it's easier and it'll drop you right at your hotel door rather than 50 feet away on the street. Good luck and have a great time!!!