Well, I can really only speak to a few of the anime shows. The main theme is usually:
Bad Digimon are doing bad things. A rag tag group of kids, along with their Digimon partners, must put an end to the shenanigans with the power of friendship.
There is continuity between Digimon Adventure 01 and 02. There have been a plethora of movies around this particular group of kids. It's great.
The third series does a reset and everything is mostly stand alone from there. Some play a bit more fast and loose with things like digivolution tracks and whatnot, but the main themes are still there. Mostly they just want to bring peace to both the "real" and Digital worlds.