Walking
This is the first and slowest form of transport you get in the game. While walking you will encounter random battles. You can't cross rivers or mountains while walking.
Chocobos
The horses of this world. There are different types of chocobos which have different abilities to cross terrain. The best chocobo is a gold one which can go anywhere on the map.
Buggy
Leaving the gold saucer for the first time you will get this buggy. You can cross shallow rivers with this. Its slightly faster then walking. You will still encounter random battles.
Tiny Bronco
Cids small airplane which can't fly. However it is good for something. It can move along rivers and shore.
Submarine
Once you have this you can start to explore the sea beds of this world. You will come across Emerald Weapon or explore the sunken Gelnika.
Highwind
The highwind is the fastest form of transportation. With it you can go nearly anywhere. You do however need flat land to land the airship. Inside there are stables for chocobos, a save point and resting area.