A few things to try:
Use "Save For Web" rather than "Export" to create JPEGs.
Check "Edit > Color Settings..." and maybe turn Color Management off.
As for Flash, PNGs will work, although Flash should be able to import Illustrator files, EPS files, and others. They may give better results as vectors stand a better chance of being retained as vectors. I don't know if Illustrator exports vector information in PNG files like Fireworks does, but I think it may rasterize it, which will limit your options.
Also keep in mind that Illustrator can export a Flash SWF file. You can also do simple animations by putting each frame on a different layer in Illustrator and selecting the appropriate options in the export dialog box (File > Export > Macromedia Flash (SWF) > Export As: AI Layers to SWF Frames)