If I recall correctly, all add-ons are loaded automatically when you start a new game or load a saved game after they are downloaded.
In order to access their content in-game, there are several different ways, one for each add-on:
Dead Money: After you step out into the Mojave Wasteland for the first time, you will receive a new radio signal on your Pip-Boy (Sierra Madre Broadcast) and a new quest (Sierra Madre Grand Opening!). Follow the quest marker to the Abandoned BoS Bunker in order to start the Dead Money quests.
Honest Hearts: The crafting recipes added by the add-on are available from the start of the game (provided level and item requirements). After you step out into the Mojave, you will receive a new radio signal (Happy Trails Caravan) and a new quest (Happy Trails Expedition). Follow the quest marker to the Northern Passage and talk to Jed Masterson to begin the Honest Hearts quests.
Old World Blues: After you step out into the Mojave Wasteland, a new radio signal appears on the Pip-Boy (Mysterious broadcast) and a new quest (Midnight Science Fiction Feature!). Follow the quest marker to the Mojave Drive-In between 11pm and 3am and activate the satellite to begin the Old World Blues quests.
Lonesome Road: After you step out into the Mojave for the first time, a new radio signal (titled only Radio Signal) appears, and a new quest (The Reunion). Follow the quest marker to the Canyon Wreckage and go into the Divide to start walking along the Lonesome Road.
Installing each of these 4 add-ons increase the max level cap in 5.
There are also two add-ons that add content, without quests, storyline or increasing the level cap:
Courier's Stash: A compilation of the 4 pre-order bonuses that the game received. Right as you start the game, you will already have all the items.
Gun Runners' Arsenal: Adds new unique weapons, ammunition and weapon mods, as well as a new perk (Mad Bomber).
Besides Courier's Stash, all the add-ons add new challenges and achievements.