The idea behind the Safari Zone is to catch many Pokémon. You'll get 30 Safari Balls to play, but you have to use them all before you walk 500 steps.
The Safari Zone is unlike other Pokémon-catching situations: you don't have to 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 Pokéball , or you can distract them with Pokéblocks.
A few things to keep in mind:
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 animals with natures similar to the Pokéblock's. An animal attracted in this way is much easier to catch.