You should stay in the HsinYi area. It's the newest area in town with the biggest concentration of shopping activities, cinemas and restaurants. You will find Taipei 101 there. At the end of SongRen Road, you will also find a nice series of mountains (albeit small) where you can do some hiking and enjoy a great view of Taipei. For cheaper rent, look for some apartments on Wuhsing Street (older buildings though).
As for your holiday resort, don't know what you want to do. If you want a beach resort, google Kenting. That's in the southern part of the island. Fr more impressive landscape, go to the Taroko Gorge near Hualien and stay at the
Taroko Silks Place. A bit pricy but worth it. The area is amazing!