Most tourists to Taiwan stay in Taipei. It's the international city, and home to the "important" museums and cultural sites. Beyond that, the country has beautiful mountains and oceans, and a lot of history if you head south to Tainan. But I made a stop about halfway down, at the city of Taichung.
I didn't spend long here - two nights, perhaps - but it proved to be a great base for exploring the central part of the island. From the classic-looking train station (seen in the first picture) you can take a ton of buses around town. I also found the taxis convenient and inexpensive (compared to US and Japan, especially). Apparently an MRT line is being built which will open sometime next year that can make access to some locations more convenient.
As you can see from the pictures, I spent an evening at a night market. Taichung has a few good ones and lots of great places to eat at the markets, so like everyone else I highly recommend you spend some time here.
As always there are a lot of clothes, and I found a bunch of character goods, shirts and other clothing/accessories especially. I don't know how much of it is actually licensed but it's fun to look at anyway.
Since Taiwan's night markets are known for the food, be sure to stop and try some of the mystery meats, delicious fresh fruits, or awesome drinks. Taichung also has a couple great museums - an art museum and a science museum.
While Taichung itself is worth about a day, it's perfect for a home base for hiking or sightseeing in the surrounding areas, and has buses to the mountains. For more information on specific locations in town, check out the Wikitravel article.