These missions are set in 2013 which is when Operation Kingfish happens, however it's unclear whether these missions actually link into the Modern Warfare story or not.
The initial training mission is CoD4's FNG. You drive in with Captain Price, do the target practice from the main game, then you do grenade training, both throwing and using the underbarrel attachment, and then you do an obstacle course. And you do not run the cargo ship course (though it appears that does get unlocked later on).
Mission 2 is set on the multiplayer map Crash. The map has been modified so that it kind of sort of matches the style of the Brazilian maps from MW2. Your helo crashes and you are captured by militia, Ghost frees you and you gather some intel. Then you have to blow up your helicopter and escape to the top of the tall building for extraction.
Mission 3 is on MW2's multiplayer map Estate (still set in Brazil though). The beginning is basically the first section of Contingency from MW2, then you have to catch a guy named Vasquez, and you're supposed to bring him in alive but someone snipes him from across a lake.