My wife's parents lived in the hill country. As much time as I've spent out there, I've only been to one museum and that was because my father-in-law had to have someone drive him three days a week for his chemo treatments. The museum of cowboy art in Kerrville. It's alright, but if you really want to show your wife some Hill country culture, the best thing about Kerrville is this guy that sets up on the side of the road as your coming into town from the east. He sells the sweetest watermelons you'll ever taste, for a buck and a half. Then go on up just south of Mason, there's a barbeque pit just south of town run by this old Mexican dude and he's got the best barbeque goat in Texas. The tortillas and green chiles at the Hill country inn ain't bad either, just don't spend the night there. Then on your way home sling some lead at Red's on the west end of Austin, or is it Buda now, anyway go there too. And if you're really interested in Texas culture, I know (from personal experience,and a few scars to prove it) of the whereabouts of about a half dozen "Texas Shot houses" but that's probably more "culture" than your looking for.
Alamo; shmalamo, go for the goat and watermelons! You'll be glad you did!