People write in different ways. I've been to talks and on courses where the advice has been to start with characters, but then again some people start with plots. I suppose different books require different starting points. My current novel One of a Kind which was shortlisted in the Waterstones WOW Factor competition and in Adventures in Fiction Apprenticeships 2006 started with a plot. I thought about a starting point, the conflicts and the one main character in detail, then I added to this: brainstorming other characters and researching historical background. That's just how I did it. That's not to say, this is the only way.
Fran, you asked about the types of plots people can use, so I'm posting a list of plots that every story apparently falls into. I got this from someone at Wordpool some time ago. I forget who it was who posted it originally, but it's an interesting set of topics.
1. Quest
2. Adventure
3. Pursuit
4. Rescue
5. Escape
6. Revenge
7. The Riddle
8. Rivalry
9. Underdog
10. Temptation
11. Metamorphosis
12. Transformation
13. Maturation
14. Love
15. Forbidden Love
16. Sacrifice
17. Discovery
18. Wretched Excess
19. Ascent
20. Descent
While we're discussing writing, let me suggest that you read a very informative blog belonging to Candy Gourlay, all about writing. It's called Notes from the Slushpile and at present she has some really good stuff on the Bologna Festival posted. I've not figured out how to add her to my links but when I do I will!
In the meantime, I'm on to my last two chapters so it's back to the drawing board.