Okay, so. I got a Pikachu a Pidgey, Ratttata and Charmander. Is there a set level that Pokemon level up to their next form?
And it's Gary.
Pokemon level up at various points through level up, mostly dependent on 'how strong' a Pokemon is. ie; Metapod evolves into Butterfree at level 10, but Dragonair doesn't evolve into Dragonite until level 55.
There are some Pokemon (like Pikachu) that level up another way, through an item, through trading (sometimes while holding an item), leveling up at day/night could effect what it turns into, and happiness. The vast majority of Pokemon you'll meet though will just evolve through level up.
There is some reason not to evolve a Pokemon right away (you can press B to cancel an evolution, and it will try to evolve at next level up). One would be something like Scyther is quite different than Scizor, and you might not want to level it up. The second would be that some pre-evolutions learn moves their evolutions will never learn, or they learn it at a much earlier level.