Unusually for a state capitol, New Mexico's state house is very accessible. No metal detectors, no lines to get through security. It also has a tremendous collection of locally themed art by local artists in a wide variety of media. Even if government buildings aren't your bag, come to see the art. On the main floor at the entrance there's a helpful information desk.
We walked to the capitol from the plaza and had some trouble finding the entrance. We were on Don Gaspar St and gave up a bit too early, winding up in the corridor connecting the capitol to the north building. Had we walked a 100 feet further we'd have clearly seen the west entrance.