Mauville City is the city in which the Hoenn Game Corner and Bike Shop are here. Additionnally, Wattson, the third Gym Leader of Hoenn, resides here.
Bike Shop- Here, you can talk to the owner and get either a Mach Bike or an Acro Bike. Also, if you want to switch bikes, you can talk to the owner again.
Mauville Gym- At the Mauville Gym, you have to get through the electric doors by defeating trainers. After you do that, you can fight Wattson, the Gym Leader. He uses Electric-typePokémon and after you beat him, you will recieve the Dynamo Badge and TM34, Shock Wave.
Rock Smash Guy's House- Here, if you talk to the man sitting at the table, you can get HM06 Rock Smash.