There are some good image editing tools linked at nonags.com. I'll leave it to others to suggest particulars, because for old images I seldom have to go outside of the Epson software that came with my Perfection 3170. It's pretty amazing what the default restoration values can do.
As far as the file size. If you really want to save "archival" quality scans of your images you arguably may wish to save them in .bmp or some lossless format, so they'd be relatively huge files, yes. But if you want "nice" quality in more manageable sizes for daily use, try this; Install freeware XnView. When you scan your images, select 300 or 600 dpi, save to jpeg with higher quality settings (90% for instance, if it lets you select by percentage), and then once your images are created, right click "open with", select "XnView", and use it to crop and/or resize them.
You'll see that your image files won't need to be terribly large. Also, with some practice, you can also do quite a bit of nice filtering using XnView.