You do need both bikes! At the same time.
I'm sure you're wondering; "W-w-what... both bikes? How could one do that? Having two bikes in one Pokémon game, that is completely outrageous." Yes - you can get both bikes - Omega Ruby and Alpha Sapphire are changing it up, completely destroying all preconceived notions of what a Pokémon game should be. Here's how to obtain both the Acro Bike and Mach Bike:
To get both bikes, you need to advertise Rydel's Bikes to various people throughout the region
•Mach Bike - Show the Mach Bike to a Hex Maniac in the Route 111 Desert
•Acro Bike - Show the Acro Bike to a Bird Keeper in the Acro Bike area of Route 119 - Requires Surf & Waterfall
•Any Bike - Speak to a Triathlete in Battle Resort
Once you have spoken to all three of these characters, return to Rydel's Cycles in Mauville City and he will give you the other bike, so you now have both bikes at the same time.
Good luck getting those TMs!