Hi there, I have been to Taipei 101 and it is quite interesting to see, sometimes though the weather might not be the best to see and can get quite hazy above... we ended up going back a few times til we just decided to go up! :) The cost was 450NT (16.00 CDN). I have yet to head to Elephant Mt., but I have heard that the views are spectacular and that if you are used to a bit of a hike it shouldn't take too long to get up. Maybe 1hr - 2hr depending how many times you stop :)
Currently if you are planning to go to the Maokong teahouses by Gondola, depending on when you are going, the gondola right now is suspended and under maintenance. You can check out the site here to see if it might be running closer to when you are going. It is a great trip up though, the views are spectacular! There is also the tea houses and hiking above as well.
Taipei Main station has the underground mall with lots of souvenior shops and clothing stores, Q-Square mall is quite pricey I find, the clothes are cute and I often like to browse through the stores but never really buy anything... Maybe if you are on the blue MRT line already check out Ximending? It's one stop away and has a lot of shops as well, plus while you're there you can also check out the Red House which is right across the street when you leave via exit 6.
You can also stop by (one station after Ximen) Longshan Temple at night which is really quite beautiful :) It's one of the older temples in Taipei and definitely worth a visit.
Also you can check out Shi-da night market which you can go to instead via the green (if you are at taipei main station) heading towards Xindian. You can get off at Guting station.
Danshui is a great place to visit for the sunset! You can walk along Old Street and head to Fisherman's wharf as well. Also on the same line (red line) you could stop on the way there to Xinbeitou and visit the hot springs if you wanted to. The stop is at Beitou and you would transfer that station to Xinbeitou.. and a lot of things there are within walking distance. At night you an also hit up the Shillin night market on the way back as it is on the same line as well.
If you want more insight of some of the places I am talking about I have written up some information and sites here: