Starting the Dawnguard Expansion
After you purchase and download the expansion when you start the game by continuing your last save you will note that the expansion is automatically part of the game -- and you will flag the Dawnguard Quest (it is called DAWNGUARD and instructs you to speak to the leader of the Dawnguard). So activate that quest so it is the only pointer and head to Riften (if you are not already there) and exit the city.
Follow the marker through the woods/down the road to the entrance to the Dayspring Canyon -- follow the path after you enter the Canyon and you will meet Agmoer, who tells you that he is a new Vampire Hunter and asks to join you. You both head to Fort Dawnguard ahead which is the Dawnguard HQ, and when you arrive you head inside to meet Isran, the leader of the Dawnguard.
There are three chat options -- for giggles tell him that you were just looking around and ask what is this place? Next ask What is the Dawnguard? He tells you -- with attitude -- and then you can ask him What can I do to help? He briefs you and you then learn from Tolon about a vampire artifact that may or may not be in a cave. Getting the briefing completes the quest DAWNGUARD and flags the quest AWAKENING.
A Chance Meeting on the Path
Isran gives you a crossbow and invites you to help yourself to supplies... Tolon and Isran argue and trade insults, and then Agmoer gets his interview -- and then Isran gives him a crossbow and starts training him right then and there in its use.
If you take the time to explore the fort you will find a bookshelf with the book The Aetherium Wars by Taron Dreth on it -- read that book to flag the Misc Quest "Investigate the Ruins of Arkagthomg." Bear in mind that while these side-quests do not really contribute to the expansion, they do provide you with opportunities to gain ability levels and thus XP by using your skills, and they are a fun and interesting addition that expands the story and intensifies your immersion in the world of Skyrim, so why not do them?
Searching barrels and crates as well as sacks is a good idea as you may find some useful resources you do not have a lot of -- and be sure to grab the 10 steel bolts off of the table in the practice range -- as having extra ammo does not hurt. Once you are sure you have looted everything that you want you may want to head back to your house in Whiterun to drop off the stuff you are carrying that you do not necessarily need at the moment, take the time to make potions there if you need to, and otherwise see to your kit.