13 quests in the Main quest.
There are 18 side quests.
In total, there is 31 quests PLUS multiple unmarked, small quests (Mostly just bringing items to a certain person or finding a stash.)
Operation: Anchorage DLC adds 4 quests.
The Pitt DLC adds 3 quests + 3 unmarked quests
Broken Steel DLC adds 3 new quests, 3 side quests and more unmarked quests.
Total amount of marked quests with DLC packs: 44 quests
The up-coming Point Lookout DLC adds atleast 3 quests.