There is a lot to travel to do in Pokémon Brilliant Diamond and Brilliant Pearl, and it will happen much faster with a bicycle. If you want to zip around the map as fast as possible, then biking is the way to do it.
To get on the bike, head to Evopoli. You will have to beat Gardenia, the boss of the gymnasium of Evisopolis, and once done you can go to the Galactic building in the north part of the city. You will also need the Cut HM, which you will get by talking to Cynthia when you first arrive in Eevopolis.
Once you arrive at the Galactic Building, you will have to fight against Team Galaxy. This fight will include Jupiter, a commander of Team Galaxy, but you shouldn't have much of a problem mopping the floor with them. Defeating them will free their prisoner and receive an invitation to Rad Rickshaw's bike shop. This is south of the Echo City Pokémon Center.
You will be able to choose which bike you want from there and you can also decide on a color. Once you have the bike you can press button B to accelerate and you can also use it to cross muddy slopes.
