Instructions
Play the game until you have beaten the first gym leader and can get to Mauville City.
Travel to Mauville City and walk to the bike shop that is next to the PokéCenter in the middle of town. It is recognizable by the collection of bikes standing outside of it.
Speak to the man standing outside the shop for him to offer you a free bike. There are two choices: the Mach Bike, which is faster but harder to control, and the Acro Bike, which is slower but can perform tricks.
Highlight the Mach Bike and press "A" to choose it. The Mach bike is the only one that is able to travel up mud slides -- a task that requires a lot of speed.
Press "Start" to open the game menu, and then select the Mach Bike. Travel to the mud slide you want to get up and line the bike up with the slide. Make sure your character is a few paces back from the slide so that he can pick up enough speed to go up the slide.
Hold down the direction button while facing the mud slide to cycle up it with the Mach Bike. If you did not have enough speed, start your approach further back.