How do I begin playing DLC?

You start the DLC by using the Fast Travel system to go to the first area of the DLC. From there, you'll receive a new mission and further instructions. These are the areas you're looking for:

• Captain Scarlett and Her Pirate's Booty: Oasis
• Mr. Torgue's Campaign of Carnage: Badass Crater of Badassitude
• Sir Hammerlock's Big Game Hunt: Hunter's Grotto
• Tina's Assault on Dragon Keep: Unassuming Docks

Note that you may be restricted from playing DLC by your level and/or story progression.

I thought DLC in the first game was handled by opening new fast-travel locations - you received the quest when travelling to them. I would assume it is the same for Borderlands 2.