I've never watched any of the Animatrix films, but I have seen the movie trilogy plenty of times. Without providing too many spoilers, how does the Animatrix series relate to the storyline of the Matrix film trilogy?
They are context movies.
It means that you won't find any direct relation (With the exception of "Kid's Story" and "The Last Flight of the Osiris") with the movies.
The main point of the Animatrix is to explain the context in which the movies lie.
I think that the movies are good to watch if you want to know more about the Matrix world, but if you are expecting something shocking or movie related, you will be disappointed.
Animatrix is an anthology of short films expanding upon the mythology of the Matrix world. Most of them act as interludes or prequels to the movie The Matrix, and were intended to be filler between the runaway success of The Matrix and the release of The Matrix Reloaded.
They're interesting if only for understanding why the machines are doing the things they're doing to the humans, but that information is later explained in some detail in The Matrix Reloaded and The Matrix Revolutions anyway.
Before the Matrix Reloaded released, WB announced a multi-media exploration of the universe. In addition to the Movie, they released a video game (Enter the Matrix), a graphic novel, and the Animatrix. The idea being that any individual element would be a complete experience on its own but that when experienced as a whole give a fuller picture.
Enter the Matrix has you play as Niobe or Ghost from the Logos and happens concurrent to the events of Matrix reloaded. In fact there are a number of scenes where cross-over happens. For example, there is a scene where you are racing a car through the city in an attempt to reach and help Morpheus. When Morpheus falls of the truck in the highway scene, it's onto the car Niobe was driving.
The Animatrix (as someone pointed out) serves as a direct leadup to Matrix Reloaded with Last Flight of the Osiris (animated by Square Enix) in which the crew of the Osiris makes the drop that Morpheus goes to retrieve at the start of the movie.
I've never read the Graphic Novel but I'm assuming it has some nice tie in to the movie as well.