the safari zone (hg ss) is where you can catch pokemon with a pokeball called a safari ball. you pay to get in and then you go wild trying to catch different pokemon. certain pokemon can be caught by you placing props and waiting for them to appear. the same is applied to pokemon dp and platinum*. one last thing, you have 30 safari balls and just like regular pokeballs they can break free if you dont get them weakened (throw mud to make them stay but they will more likely break free of the safari balls wasteing 1 ball. bait will make them want to run but be easier to catch)
Safari Zones also have a step limit. Usually, it's 500 steps before you're kicked out. To circumvent this, just lightly tap the dpad so you turn, but stay in place. The Pokemon you can catch vary by location in the Safari Zone, so you'll need to visit multiple times to get them all.