Good choice.

Taiwan is a terrific place to travel and you can do it on a tight budget in a lot nicer way than most other modern countries, not to mention there's really a lot to see within a very short distance. What I suggest is picking up a Taiwan Lonely Planet somewhere and browsing through. Taiwan is wonderful in that you can tour it doing a circle around the island, go from north, to east coast nature, down to semi-Thailand Kending beach chillout, and come up through west coast urban and culture delights.

Taiwan is trying to encourage youth travel so there are a lot of promotions. Check out this website : http://www.youthtravel.tw/youthtravel/indexMain.jsp (switch to English).

If you're really low on the budget, I strongly recommend Coachsurfing (http://www.couchsurfing.com, http://www.hospitalityclub.org/), but the hostels around Taiwan are pretty good, although not that cheap. The option I sometimes use is love hotels, which is a lot more decent than it sounds, and is great value for money. The way I usually travel - bus/train between destinations and then renting a cheap bike for a day or two (200-400NT$ a day, around 5-12US$). Just make sure you have an international driving license for a scooter.

3-4 weeks should be great, and November is relatively good weather, although it could get a bit cold, so bring some warmer cloth as well. Even if you don't bring things along, things in Taiwan are way cheaper than Europe and North America, especially clothing, food and electronics.

You can also browse this website for more information.

Good luck, let me know if you need anything else.