Q: Car starts, but car won't go into gear

If in fact the headlights and the shifting issue were a result of changing the battery, the only circumstance that I can think of would be a voltage spike. Was the ignition key on when you connected the battery? Did you accidentally put the battery in the wrong way? Did this happen immediately or after the car was running? Alternators that are producing too much voltage can destroy a battery, blow fuses, and burn out bulbs quickly, headlights usually blow first. There is a electric solenoid that releases the shifter when you apply the brake. Somewhere on the shifter bezel there is a plastic tab you can remove and take a small screwdriver or other tool to manually release the shifter lock. Check all your fuses or have a mechanic inspect and repair your car’s headlight problem.

As for the gear shifting issue, this can potentially be a separate problem with many potential causes. If you need help, a qualified mechanic from YourMechanic would be able to come out and diagnose your car’s gear shifting issues.