You mean what stage in the game? I'm nearly positive it's toward the end of the game. And since that game is literally twice as long as any other... plus you have to do that minigame thing.

In Diamond, Pearl, and Platinum, you can't use Pal Park until you've beaten the Pokémon League. And in HG/SS, Pal Park takes the place of the Kanto Safari Zone, if I remember correctly. So either way you won't get to use it until the postgame.

If you mean actually catching the Pokemon, not too terribly long in my experience, you run around a bit and tend to capture them quickly. If you're just transferring one team, thirty minutes, tops, if you're transferring a couple boxes... give it a few hours.