How much to compress?

To best preserve image quality, the best compression is "lossless" compression, such as PNG (Portable Network Graphics) format.  Lossless compression means that no information is lost when the image is decompressed.  That means that you always see the original image as it was before being compressed.

My camera stores pictures in JPG (which is lossy) so I store them on my computer in JPG, BUT if I edit the image (crop, redeye, etc) then I save the image in PNG.

If you re-save a JPG, GIF or other lossy compression, your image will lose information every time you save it.  Therefore, other than uploading from your camera, you should always save in PNG format.

I share in the following view of many other professional photographers/digital imaging professionals...

If you shoot in RAW, save your RAW files as well as your "converted to JPG" files.  Choose level 10 quality duting the conversion.

If you shoot in JPG, always save your original files, then save your edited files as level 10 JPG.  Should you need to make another version or edit, do so from the original file, not the already edited one.  Never save over the original files,  Instead, rename edited versions.

Saving in JPG is fine, just use the highest quality level (lowest compression level).

Saving to a lossless/uncompressed file format will NOT improve quality to a decernable level if you folow the above methods.  It will however cause your files to be unnecessarily large.