I usually start by drawing a map. Then I flesh out the geopolitical makeup of one region on the map (government types, religions, etc.) Then I usually role up some influential NPCs based off of that. They, in turn, almost always spark a few adventure posibilities, which help me focus on an even smaller portion of the map to flesh out fully. Then for fun I make one more villian who could shake things up, or be a red herring to whatever else is going on.
For inspiration, I agree that it comes from just about anywhere. Sometimes it's a matter of wanting to do something that you haven't seen anywhere else out there that drives a homebrew campaign. For example, until the Savage Tide adventure path started in Dungeon, I was hoping to run something with a decided ancient South American feel to it... sort of the Incans, Mayans and Aztecs all thrown in close to each other.
As for tools, good ol' pencil and graph paper for me. I know there are some better tools out there. I just don't have the moolah to bring 'em home with me.
"I never apologize, Lisa. I'm sorry, but that's just the way I am." -Homer Simpson