Win,
I went F&E. Just book taxis for most of the days. 1 or 2 days we went around city by MRT. Depends on how many days you have, you can plan say 2-3 nights in Taipei, 2 nights in Yilan and 2 nights in Hualien. Skip Hualien if you don't have enough days. Yilan has many nice minsu facing paddyfields, very pretty. Hualien - can book taxi to visit the toroko gorge / hualien ocean park / farm / night market. Yilan - there are some cultural museum and hot / cold spring which kids can also play. Yilan and Hualien are along east coast of Taipei. TP -> Yilan (1.5 hrs) -> Hualien (2 hr) by fast train. The fast train is very comfortable and easy to book. Once you book the minsu / hotel, they can pick you up at the train station, arrange for cab driver and also buy train ticket back to TP. Alternatively, can go Taipei and Cingjing (near Taichung). Cingjing is farm in the mountain area. Can feed the lamb and go near to the sheeps and cows. I prefer to go F&E so that can choose those place that we want to go. With children, if you join tour group, half the time will be on the road or rush from one point to another.