gamefaqs.com will tell you all you want to know about capturing.

You can cross the entire length of Spira on foot and by boat (I've tried), but the Airship lets you travel to any region instantly.

You'll want to capture when you set aside time specifically for capturing. The best time is once you open up the dungeon in (SPOILER)Sin, since it's the last dungeon that contains monsters that can be captured.

You'll get a lot of some particular item for capturing at least one of each monster in a given area. It will also unlock a specific monster that appears only in the Arena.

You'll also get 99 Dark Matter (!) for capturing five of every monster in every area. They are the best customizing item.

You'll get 10 Master Spheres for capturing ten of every monster in every area. They let you move to any point on the Sphere grid. Nifty, but I would've liked more Dark Matter.

You'll also get items for capturing a certain number of every species, but if you go for ten of every monster in every area, you'll get those anyways.