So you need to strike a balance between quality and file size. It's only at the lower end of the quality range that the human eye can actually perceive any difference. So you can usually decrease the quality quite a lot before the picture degrades too much. In fact, you will almost never need to save a graphic at the highest quality because even a medium quality should be sufficient. Scan the final version at 300 dpi.
Re-size the picture to a reasonable size.
If it asks you to choose a compression factor or a quality level before saving, choose a medium level. DON'T choose high quality - the file will take too long to download. There's a very easy way to save any picture you see on a Web page to your computer. A menu will pop up. This technique applies to any Web site you visit.