Take it from a real coward: Taiwan is a really safe place to travel. The worse dangers are snakes -in trails, mountains- getting lost in the wilderness, and food poisoning -from not fresh places. There are many interesting destinations you can reach by train, which is easy to take even if your Chinese is not that great, cheap and fast. Fulong or any beach on the East coast, Jiufen -take train to Ruifang-, Sanyi or Yingge are one day trips that are really nice, close to Taipei for starters. If you like cats, go to Houtong. take a bus to Yangming Shan, or Daxi or Beipu. Look these places up on the 'net or the books recommended. Bring a camera, take lots of pictures, even by yourself you'll have lots of fun.
As to transportation, to err on the too cautious side, remember to take taxis from reputable companies, like Taiwan Taxi or Metro Taxi. Better avoid independents as they are not that clean -they do not have to adhere to company standards- and older cars too. I usually avoid taxis with any dents.