Mauville City can be reached from Routes 110, 111, 117 or 118. Once there, visit the Bike Shop. Fortunately, it's not hard to find -- it's the white building with the red roof near the north end of town that has a yard full of bikes next to it.
When you first visit the Bike Shop, the proprietor, named Rydel, offers you either the Mach Bike or Acro Bike free of charge. The Acro Bike is needed in the Safari Zone for its jumping ability. If you've already been to the Bike Shop and chose the Mach Bike, it can be traded in at any time for the Acro Bike at no charge.
The white bridges in the northeast area of the Safari Zone can now be crossed. The trick to moving between bridges is to perform a normal jump by pressing the B button and the direction you want to move in simultaneously, rather than performing the bunny hop.