Devil May Cry Storyline *no spoilers*?

I just started on the Devil May Cry series only because 4 came out and I realized I was SERIOUSLY behind. When I bought 1, the guy said that Devil May Cry 2 was complete bs and had nothing to do with the storyline. Now I'm hearing that DMC 3 follows directly after DMC 1 and 2 is the precursor to 4. If that makes any sense to anyone who already played or knows the Devil May Cry 4 story line, could you tell me if that is true? And if not, what order do the games go according to the actual story.