Mauville City is the city in which the Hoenn Game Corner and Bike Shop are here. Additionally, Wattson, the third Gym Leader of Hoenn, resides here.
Poké Mart- you can buy TMs here, talk to the second clerk
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 receive 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.
ALPHA SAPPHIRE AND OMEGA RUBY
Mauville Food Court- Battle Trainers on the right turn to get a nugget daily.
Reflexology Place- Makes your leader Pokemon more friendly to you daily.
Battle Institute- Here you can try a Battle Test if you have beaten the Champion Steven Stone.
Inverse Battle Stop- Here you can battle a Trainer with the type matchup backwards.
Ultimate Move Studio- Here you can teach Pokemon moves like Blast Burn and Frenzy Plant.
Pledge Move Dojo- Here you can teach your Pokemon a Pledge move (Fire Pledge, Water Pledge, Grass Pledge). They are useful for Double and Triple Battles.
Crooner's Cafe- Talk to the bald guy to get your Keldeo or Meloetta to learn a move.
Mauville Game Corner- Talk to the bald guy in front to get a Torchic, Mudkip, and Treecko Doll (starter doesn't matter)
PokeMileage Center- Trade PokeMiles for Prizes.
Rizzy Ribbon Retail- Buy Ribbons for your Pokemon.