I've been through Des Moines several times, but really hadn't looked around much.
The Iowa History Museum is just west of the state capitol building, and is in a pretty large building of its own. They've done a good job at covering the broad history of Iowa, although I would wish that some of the lights weren't turned so low that it was hard to see where you were walking, let alone being able to read some of the (even darker) printed information.
One of the things I hadn't been so aware of is the extent that Iowa was once wetland. The draining of these marshes was a big part of the settling of the state. It highlights that much of what you see in Iowa, today, is far different from what it was two hundred years ago--lots of farmers worked very hard to make it that way.
The Des Moines Art Center is one of the best art museums in the region. They've done a nice job of focusing on quality. You'll find many names that are familiar even if the particular works on display are not the ones that the artists are famous for (those will be hanging in bigger cities than Des Moines).
The Iowa State Capitol Building isn't much different than many other state capitol buildings: everybody wants to copy the nation's capitol. But, Iowa has done a really nice job at both restoration and preservation (thereby avoiding the drop-ceilings and awful remodeling that was all part of the 1960s).
Looking west, over Des Moines, from the capitol steps.
The Law Library
The office of the Secretary of State and the Iowa constitution
Executive branch - the Governor
Judicial branch - the Supreme Court
Legislative branch - the House
Legislative branch - the Senate