If  you come to Fort Worth in January you can go to the Fort Worth Stock Show & Rodeo.  Three weeks of rodeo, livestock competitions, Texas Food, even a carnival.   Rodeo is 2x a day at the Stock Show; it is part of the Rodeo Circuit, so the competitors earn points toward the National Championship.

 Right next door to the Stock Show are 5 museums:  Kimbell Art Museum, Modern Art Museum of Fort Worth of Fort Worth, Amon Carter Museum of American Art, National Cowgirl Museum and Hall of Fame and Fort Worth Museum of Science and History.   

South of this complex on University Drive you will find the Botanical Research Institute of Texas (BRIT), Fort Worth Botanic Garden and The Fort Worth Zoo.  

 So you can do the Wild West thing at the Stock Show, soak up the culture at the various other places, and then go downtown to Sundance Square and enjoy the nation's best downtown, or go to the 7th Street  area between the cultural district and downtown.  

I actually answered a similar question recently. Check out the Fort Worth Stock Show & Rodeo and Mesquite Championship Rodeo -- both are great!