B.G.
Yes, definitely the Lincoln Presidential Museum. There are several Lincoln sites. You can visit his home, his tomb, the Old State Capitol, Lincoln's Law office. New Salem is just north of Springfield. It is where Lincoln lived before moving to Springfield.
There is a tourist office on the south side of the Old State Capitol (at least there was a few years ago) where you can get a map of the downtown area. There are a lot of unique restaurants within walking distance of both the Lincoln Museum and the Old State Capitol.
The Illinois State Museum is pretty good. There's some natural history exhibits and some about the history of Illinois.
If you were looking for something other than Lincoln, check out the calendar in the Illinois Times:
http://www.illinoistimes.com/Springfield/events
They do a nice job of listing events each week. You might find something your kids will really like there.
Feel free to send me a message if you have any more questions about Springfield. Hope they have a great time.