The idea behind the Safari Zone is to catch rare kinds of Pokémon found nowhere else in Hoenn. You'll get 30 Safari Balls to play, and get teleported back to the Safari Zone entrance once all 30 are used or once you walk 500 steps (whichever comes first).
The Safari Zone is unlike other Pokémon-catching situations: you don't actually battle any of the creatures that you meet. Instead of that, you have two options. Either you can creep closer and attempt to use the Poké Ball, or you can distract them with Pokéblocks.
Below are some tips to help you in your catches:
Getting closer to a Pokémon increases your shot. Don't move too close, or you could frighten it away.
Use the Pokéblock Feeders. Pokémon around the feeders are more likely to be caught because they're distracted.
Specific Pokéblocks attract Pokémon with natures similar to the Pokéblock's. A Pokémon attracted in this way is much easier to catch.
One way to save steps is to stand on a square in the grass and flick the D-Pad in a different direction or use Sweet Scent. You won't take a step. Additionally, jumping in place with the Acro bike can accomplish the same.
Use the Mach Bike to go up the cliff to the west, then to the northern pond. Cross it to find TM22 (Solarbeam).
Use the Acro Bike to go across a bridge that needs to be jumped. Phanpy can be found in this region of the Safari Zone, and there is also a Calcium.