Birmingham and Salt Lake City are about 1,463 miles apart by road. Split the difference and you arrive near Oklahoma City, the city closest to the halfway point between them. That puts roughly 13 hr 5 min of driving on the Birmingham side and 18 hr 4 min on the Salt Lake City side — the fairest single meeting point among the cities near the middle.
Over this distance most people will fly rather than drive the whole way. Oklahoma City still makes a fair, central place for Birmingham and Salt Lake City to converge, splitting the travel instead of asking one side to cross the country.
If Oklahoma City doesn't have what you're after, Kansas City and Dallas are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Birmingham and Salt Lake City.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.