For those who have a bent towards western heritage, here are a few that I have been to and recommend. Many more are out there for sure. Listed by city in no particular order.
Tulsa - Gilcrease Museum
Oklahoma City - National Cowboy & Western Heritage Museum
Lubbock - National Ranching Heritage Center, Texas Tech, Museum of Texas Tech University
Colorado Springs - Pro Rodeo Hall of Fame Museum of the American Cowboy
Fort Sumner, NM - Billy the Kid Museum
Fort Worth - Stockyards Museum, National Cowgirl Museum & HOF, National Multicultural
Western Heritage Museum, Texas Cowboy HOF
Waco - Texas Ranger HOF and Museum
Gene Autry, Ok - Autry Museum of the American West
Corpus Christ/Kingsville, Tx - King Ranch Museum in town and tour of King Ranch proper just
outside of town. While in the Kingsville/Corpus Christi area - not western
heritage but well worth a visit - USS Lexington Aircraft Carrier Museum with a
caveat - visit the Lex in the cooler months. All that metal and the flight deck
get very hot in the summer months.
Rockport, Tx - Fulton Mansion
Adding to this list, I recommend The Museum of the Big Bend in Alpine TX.