There are 4 creatures you can ride in Golden Axe III. The fastest way to get this is to go UP at the first path choice then take the UPPER path at the second path choice while riding all the creatures you encounter along the way. You'll ride your 4th creature in the fifth level this way.
The 1st creature a short tongue snail can be found in the very first level. At the end of this level take the path UP.
The path up leads to the second level but it has no creatures to ride so continue to the end.
The third level has the 2nd creature a long tongue snail. At the end of this level take the UPPER path again.
The upper path leads to the forth level which has the 3rd creature a red fire-breathing lizard.
The fifth level has the final creature you need to ride for the achievement. A green lizard that throws enemies across the screen when you attack.