You need a car in Alabama. Public transit exists in Birmingham and a few other cities, but it is limited and will not get you to most of the places worth visiting. The state is spread out and the best scenery, parks, and small towns are off the interstates.
Interstate 65 is the main north-south artery, running from the Tennessee border through Huntsville, Birmingham, Montgomery, and down to Mobile. Interstate 20 crosses east-west through Birmingham. US highways and state routes connect the smaller towns and are generally well maintained.
Driving distances are manageable. Birmingham to Montgomery is about 90 minutes. Birmingham to Huntsville is around an hour and a half. Montgomery to Mobile is two and a half hours. The Gulf Coast beaches are about three hours south of Birmingham.
Car rental is available at all major airports. Gas is cheap by national standards. Traffic is light outside of Birmingham's rush hour.