I second the Nature Center. There are no trains but I've heard this is pretty cool - http://www.lazoomtours.com/. If you're into hiking at all there are some nice trails at the State Arboreteum or UNCA to mess around on. Actually there's a trail at the Nature Center too now that I think about it. If you're around Friday night, kids (of all ages!) like the drum circle at Pritchard Park - it's right in downtown, anyone can tell you where it is.
There are also some great playgrounds - one on Azalea Rd about a mile past the Nature Center, one at Carrier Park.
Another thing to keep in mind is that Mondays are kinda quiet downtown. I think the Health Adventure is closed on Mondays and a lot of restaurants are too. Checking the MountainXpress is a good idea, like already suggested.
Enjoy your trip!
We just got back home today and we had a really wonderful time! :
Thank you for all the suggestions!
We went to the Nature Center, went apple picking, went to the Health Adventure, went to Littlest Birds and Nest Organics, and otherwise just had a great time wandering around downtown and popping into great shops and eating some wonderful food. Wow, the chocolate at French Broad Chocolate Lounge is amazing!! Between Nest Organics and chocolate, I was in heaven!