Being from Illinois, I always joke that it's state law that school children have to visit the Lincoln sites (anyone remember when they were called Lincoln shrines?) several times throughout their school lives. I can't count how many times I've been to Springfield, but it seems I always come back with something new.
I enjoyed the Lincoln museum with the exception of the election of 1860 part. Tim Russert doing it like a newscast was far too cheesy for me. Every other aspect of it, however, was wonderful. The funeral exhibit was powerful and moving. I even liked the food (although I felt that the General McClellan chicken salad sandwich was a bit of editorializing
Additionally, there are several other sites outside of Lincoln which deserve mention. The Dana-Thomas house is a shining example of early Frank Lloyd Wright architecture. Also, the Vachel Lindsey house tells the sad story of the poet who wrote "Abraham Lincoln Walks at Midnight" but couldn't extricate himself from ill health or bad debts and died by drinking a bottle of Lysol. Prairie Archives is my favorite used book store and is located on the Old State Capitol Plaza. Just two doors down is the Feed Store, which sells the best soup and sandwiches known to man.
Springfield is a wonderful place to visit and anyone who goes there should also make time to visit two other sites within driving distance--New Salem (20 miles northwest of Springfield) is a recreation of the village where Lincoln lived and, possibly, loved (if the Ann Rutledge story is to be believed); and The Lincoln Log Cabin historic site near Charleston, where Lincoln's father and step-mother are buried.