Those using Linux can install and use Kim (KDE Image Manager), which has about 40 options on the Right-Click menu.
For reducing file-size while retaining good image display, use the Actions > Compress and Resize > WebExport, option. Defaults are 640 x 450, 800 x 600, and 1024 x 768, which covers most Forum-etc maximum sizes. Those are for 4:3 AR images. If using a P&S in 3:2 AR option mode, or a DSLR, the width quoted is retained, the height changes.
For a 4MB std JPEG at 3648 x 2736 4:3 AR original, the file-size will reduce to 194kb at 1024 x 768, or 128kb at 800 x 600.
To further reduce the file-size at 1024 x 768 use one of the default re-size options to reduce the size to say, 1400 x 1050, then the WebExport function, to 1024 x 768. On the 4MB original camera file mentioned above, this makes the WebExport file 147kb at 1024 x 768.
As that is two operations on the lossy JPEG format, some degradation must occur, but it isn't noticeable at the online display size.
A better option is to do the display size reduction, then, on the same drop-sub-menu, use Compress Custom, to get just below the max file-size allowed, while retaining most quality.
You can also do a Custom Re-Size, between the 6 offered re-sizing defaults (or smaller/larger) - then do a Compress Custom to suit that, to the desired file-size.
(Kim is part of the free Linux KDE desktop environment, but will work in other enviros, such as Gnome - some might require you to install KDE-base, which won't affect other functions. Kim can add a border to images, add on-image text to images, losslessly rotate images in 90-degree increments, or Custom angles, while retaining the Exiif, rename files, create a montage, slideshow or gallery, and many other functions. Quite useful for a 115kb install..!)