For breakfast or lunch, Patachou (several locations), Taste, or Zest would be good. The Illinois St. Food Emporium isn't a bad sandwich shop either. If you'd rather go for grit and character, have your friend pick his favorite pork tenderloin sandwich place.
For dinner, you can't go wrong with L'Explorateur, Elements, or R Bistro. Scholar's Inn isn't bad either. The latter three are all downtown.
St. Elmo is second rate for steaks I think, but it is definitely worth grabbing a drink in the bar and having a shrimp cocktail.
The Eiteljorg Museum (the American West museum you mentioned) is actually pretty good if you are into that sort of thing. Both excellent traditional western art, contemporary western art, and North American Indian artifacts from all major regions. It is the best museum of its type I'm aware of east of the Mississippi.
If you haven't been to Indianapolis, a walking tour of downtown would be in order. My tour starts at the Artsgarden, a glass wintergarden suspended over the intersection of Washington and Illinois. Walk west o Washington to see the Indiana Theater, a terra cotta Spanish-Moorish movie palace. Turn north on Capitol Ave. to see the Indiana State House. It is definitely worth going inside. Turn east on Market St. and head to Monument Circle, one the best urban spaces in America. There is obviously the stunning Soldiers and Sailors Monument, as well as the Circle Theater (another movie palace and home to the Indianapolis Symphony), the art deco Circle Tower, the Columbia Club, and the 1857 Christ Church Cathedral. Turn north on Meridian St. Come to the Federal Court House (worth a view of the interior, though you need to go through metal detector). This is the first of a seven block neo-classical memorial mall that is also one of America's great urban spaces. North of the court house is University Park, followed by the imposing Indiana World War Memorial (the interior, both downstairs and the shrine room upstairs are absolutely stunning, but closed Monday), followed by Veteran Memorial Plaza (home to the obelisk and memorials to Korea, Vietnam and WWII, followed by the two block American Legion Mall, followed by the Central Library (both the old neo-classical main building, and a new modern addition). Flanking the mall on the west is the Scottish Rite Cathedral. On the east is the brutalist federal building, which was semi-cloned from Boston City Hall. This should keep you busy
If you drive down Mass Ave., be sure to check out the Eygptian style Murat Temple. You'll definitely see the massive new Lucas Oil Stadium on the way into downtown.
I hope this helps.