In addition to having the option to have one of your Party Pokemon Follow you outside of their PokeBall, there are also a number of specific Pokemon that you will be able to ride upon when using the same process. These are usually larger Pokemon that can support the weight of your trainer, and they will also gain extra affection towards your trainer when ridden upon.
Some of these Pokemon can only be ridden when surfing around waters using the Sea Skim Secret Technique, while others will actually allow your trainer to travel faster through the area they are in. After defeating the Elite Four, you will even be able to fly up in the skies above routes and cities by riding certain Pokemon. When riding an airborne Pokemon, not only will you encounter certain Flying Wild Pokemon, but you will also be able to bypass grounded trainers and obstacles. (You will not, however, be able to fly over gates or from one route to another.)
Rideable Pokemon
Charizard: Flying In The Sky
Persian: Increased Movement Speed
Arcanine: Increased Movement Speed
Machamp: No Special Effects
Rapidash: Increased Movement Speed
Dodrio: No Special Effects
Haunter: No Special Effects
Onix: Gives Access To Sky Wild Pokemon Encounters
Rhyhorn: Increased Movement Speed
Rhydon: No Special Effects
Kangaskhan: No Special Effects
Starmie: No Special Effects
Tauros: Increased Movement Speed
Gyarados: Only Accessible When Using Sea Skim
Lapras: Only Accessible When Using Sea Skim
Aerodactyl: Flying In The Sky
Snorlax: No Special Effects
Dragonite: Flying In The Sky