In the spring there are a couple short(ish) drives from Des Moines with great bike trails - Bentonville/Fayetteville Ark has a great bike trail and would be about a 6 hour drive. Tulsa has enough trails to keep you busy for 3 days and is also about a 6 hour drive. Mississippi with the Natchez Trace would be a little farther but probably worth the drive. (I've never been to the Natchez Trace in Mississippi but it's on my bucket list).
In the summer, as long as you are OK with limestone trails look no further than the surrounding states. Plenty of trails you can ride for days. Katy Trail in Missouri, Cowboy Trail in Nebraska, Mickelson trail in SD, Sparta Trail in Wisconsin. Also, you mentioned Minneapolis but the best trail in Minnesota is easily the Root River trail in the southeastern part of the state.
Living in Minnesota I may be spoiled with the trail miles available but as a native Iowan and veteran of many of Des Moines area trails I tell everyone that will listen that Des Moines has the best and highest quality trails in the Midwest.