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Illustrator can work with tif, jpeg, psd among some other formats. Don't make the file too big. Illy pops up a warning when it's bigger than 1500 Pixels anyway
First I would try and lighten the background in Photoshop so it's white and delete the dark spots in the edges of the scan.
In Illy you don't just hit the trace button, but click on the little arrow on the right that leads you to the tracing options. Do you want to trace colored or Black and white?
For black and white traces you can choose between fills and strokes. If you want to capture the drawing gestus, fills will be suitable. If you like to achieve exactly tha same line weight for all of the figures later, choose strokes. Illy will then trace single lines in the middle of the pen lines (or whatever tool those figures were drawn with).
Check the "Ignore White" option (Illy CS3 and up), so that the background won't be created as dozens of white objects.
Then check "preview" and adjust the other options. If that's the biggest scan you have, you will have to choose a rather small amount of "minimum area" (below 25) and "path fitting".
For both applies: the smaller the value, the exacter the result, but the more anchors in paths
Then you can only try and work with the "corner angle" setting.
After doing the trace, you have to expand it (in order to use the paths).
There's a quite detailed tutorial available at: http://www.adobe.com/products/illustrator/pdfs/creating_vector_content.pdf
Thanks so much. My image is monochrome blue. I think I may have enough to proceed. I'll let you know. I'm using CS4 at the moment so I'll learn how it differs from CS2 in the tutorial.
In CS4 you have the additional option to "ignore white" and that's all of the difference - at least all that happens on the surface.