If your saving a working version with the intention that you want to go back and edit it some more like a master copy then always save in a lossless format whether that is compressed or not. That could be BMP (not compressed) or PNG (compressed but still lossless). Personally I would use whatever is the main editing app format so that it can preserve layers and so on as well. Dont worry about file size.
When exporting a copy for people to use / look at but not edit any further - like on here - then flatten all the layers and save with either a lossless compressed format like PNG if it has detail that needs to stay clean like lines and stuff or use something like JPG which is compressed and lossy. You can specify how lossy you want it and that will determine the file size. More lossy, more loss of detail but smaller file size. Save with something like 80% and it usually looks great but cuts the file size down a lot. There are more tips but thats the basics. You can slightly blur it which will help a lot, align grid like details on 8 pixel boundaries, not have lots of vertical lines in PNG and adjust the colours so that they fall into discreet zones all helps too but all of these are for when you need it real small.
Another tip is that both PNG and JPG have greyscale options too. So if the file is black and white only then make sure you use them and it will cut it to about 1/3 of what it would otherwise be.
Researching file formats? That way lies madness. Thank you for all that. I shall convert them to JPEG and upload them again to the appropriate area of the forum.
JPG is ideal for posting online as long as you only save it once. So as Redrobes explained save your file in your software's native format (whatever that is), but export your file to JPG once you're ready to share it. A very simple explanation on how JPG compression works. When you save a JPG file, every 8 x 8 pixels are averaged in color so the color gamut is less, but the file maintains most of its correct coloring. However, if you make an edit (in GIMP/Photoshop) and save that same JPG file, it compresses again (every 8 x 8 pixels are averaged in color), so that things start getting lossy - pixels are missing, color gamut is shrinking. Do an experiment create a JPG, then save, close it, open it, save it (save it a half a dozen times this way) and your image turns into a blur with detail and color going away. Never save a JPG more than one time. However, because the file size is so small, even if you use a higher compression amount, as long as you only saved it once, JPG is compact and has decent image results.