American presidential elections

I've just started my CJ runs for president fic for femgenficathon, and so I've been reading up on the American presidential election process. Before I screw it up horribly in the fic, can someone confirm that I've understood it correctly please:

1. People decide they want to stand and run around campaigning in various states. Then each party has a national convention, where the states cast their votes for who they want as their presidential candidate. At this point, each candidate already has a vice-president, who may or may not have been campaigning with them (I think - this is where I'm not so clear).

2. Having chosen their candidates for president and vice-president, both parties go off and campaign and there are a series of primary elections which serve... I don't know what purpose.

3. There's a national election and someone is declared the winner.

That's about as much detail as I think I probably need, but can someone please tell me if I've got myself horribly confused?