You should really go to the Aboriginal Culture Center or museum thingy...it's well worth a visit and probably the most informative museum in Taiwan, beating the National Palace museum hands down in my book.
There is a night market. One is quite small and held on the street outside Carrefour on a Saturday evening. The other is bigger and is held on a Sunday evening, but I can't remember the name of the street it is held on, although it is a big road with white and yellow lines painted on it and I believe there are some traffic lights stationed along it somewhere. It's not too bad as far as night markets go. I'll get back to you with the name of the street.
There is an American style diner called Hokie Diner which sells food that originated in Germany, like hamburgers. The food is really not that bad. It is run by an ex US marine and his wife, who are both very decent indeed. I'll get the address for you also soon.
Then there is an American style Pizza place which sells food that originated in Italy, like Pizzas. These are really nice and the owner, who I believe is from Tennessee, can play the banjo very authentically. It's a shame he can't sing. A really nice guy - his name is Pete.
And then there is the Indian restaurant, which sells food which originated in India, like curry. Kasi, the owner from Pakistan, is very amicable if I may say so myself. His food is wonderful and pretty much what you would expect to find in a restaurant in Birmingham, UK. I can't recommend it highly enough.
Kasa is a restaurant and coffee shop which is half indoors and half outdoors. Interesting when it rains, but the whole atmosphere is pleasant with ambient lighting. The food is pretty good and so is the chit-chat with the various guests that turn up. Well worth a visit.
For things to do in the city, Taidong old train station is quite interesting if you want to have a wonder around. Kids always like trains, and there is an old train and turntable as well as some old Japanese yard sheds and signals you can have a look at and waste half an hour with. If you like cycling, you can rent some bicycles and ride along the old railway track for several kilometers through the city and into the countryside - the old track is now a cycle path and is peppered with quite a few coffee shops.
You can drive north of Taidong to Dulan if you like. This is a place where flaky foreigners hang out to paint and compose poetry, while wearing sandals and occasionally going surfing. There is an old sugar factory there and it is interesting to go in and have a look at all the old machines that are still in there, left in situation. Within the compounds of the factory, there are some coffee shops and art outlets selling paintings and postcards and handicrafts like driftwood furniture. It's an interesting enough place to spend an hour or two. Also in Dulan are two historic sites, clearly signposted, which date back to the days of Stonehenge, if you are interested in that kind of thing.
As Guy in Taiwan noted, Luye is great for families. The raised terrace area has some pleasant coffee and tea shops, and there is a nice view which expands across a valley where you can watch the para-gliders launching, sometimes unsuccessfully. On the weekend at the moment, there are tethered balloon flights. Aside from being $500NT each for a 3 or 4 minute "flight," I wouldn't recommend getting in any of the baskets. As an ex-balloon pilot myself, how they are running the operation at the moment seems very un-professional and slightly dangerous, so stay away from the balloons. However, they are a pleasure to see and photograph, and there are stalls selling food there and other things to for tourists to do if you are so inclined.
Alternatively, you can go down to the landing area in LuYe and watch the para-gliders land and pack up their gear. On a weekend though, hoards of bicyclists come round and park up their bicycles. Then, inexplicably, they all lie down and look at the sky while a tour guide with a loud speaker shouts at them and informs them they are looking at some clouds. You can enroll on one of these bike tours yourself if you so wish, and I suppose they are good activities for families as long as you don't mind the relative tranquility of the area being turned by a megaphone into what sounds like the immediate vicinity of a busy Watson's, on a Saturday afternoon.
I could continue, but perhaps you could tell me if you have your own transport or not.
Hokie Diner: 141 Fu Jian Road 0987 959301
Uncle Pete's Pizza: Section 1, Lin Hai Road, #143. 0952 179165
Indian restaurant: Outside Watson's in Taidong City. Just find Watson's (there's only one) and cross the road. It's there.
Good Thai Restaurant: 732, Xinsheng Road. 089 239489 (No English)
Kasa coffee: 102 Heping St., Taidong City, Taidong 0970 839354