I hope you're finding some interesting ideas for Taipei.

A couple things I really like going to the night markets. I think Shilin is my favorite. See: http://en.wikipedia.org/wiki/Shilin_Night_Market

You can have a great time just walking around looking at the food vendors, people watching and such. And if you're adventurous even try some of the interesting food.

Speaking of food, one of the most famous restaurants there is Din Tai Fung. They are famous for their steamed buns (I call them squirty buns). See: http://en.wikipedia.org/wiki/Din_Tai_Fung

There is an amazing bookstore chain called Eslite. It is way more than a bookstore having lots of other shops inside. (Ask the hotel people how to get to the newest Eslite. It is really a shopping center with books and all sorts of boutique shops)

Just find local areas around your hotel to explore. There are lots of street markets where locals go to shop for food and other goods. Very fun to explore.

In my experience Taipei is very safe and clean.

Have fun!