When I was staying there, my friends and I used to walk into 7-11 during the week, pick up one of their local travel guidebooks, and then travel on the weekends.
You don't have to travel very far actually, around the Greater Taipei Area, like the Yang Ming Shan nature reserve, but you might have to travel quite a bit inside to get to the less crowded places. Then there is a nature trail at Nei Dong forest reserve at Wulai, a bit further will be Yingge at Sanxia. There are a few nice places in Elan County just a few train stops away (Can't remember the names). Down the same train line to Hualien, the most famous will be the Taroko gorge, or you can continue into the Huadong Valley towards Taidong.