You should have the quest Dawnguard go do that to start the dlc's main story and before you actually join the Dawnguard fully you'll get the offer to join the Volkihar instead.
If you don't have the quest when your about level ten guards should start talking about it and Durak the recruiter should show up, but you can just go to Dayspring Canyon and Fort Dawnguard without it.
Yes if you want to be a vampire become a vampire free Serana and take her home. You meet her father he gives you the choice to join his court and grants you the power of the vampire Lord. The only way you end up with the Dawnguard if you bring her to the castle. I personally have never gone that route with that quest. Those are the only quest I have never done.