This will be of little value to anyone using Paint Shop Pro or a similar high-end graphics program, but is intended for anyone accustomed to using a basic program, such as MS Paint - feel free to shudder, but read on. All that's required is your paint program (MS Paint or freeware Pixia, which is a very decent alternative), plus MS Powerpoint.
First, create your image quite a bit more oversized than required. The more fine the detail, the larger it ought to be. Take an airline logo or military roundels for example; create these in a large enough size that the "jaggies" found at smaller resolution are not noticeable when the image is zoomed out from. Save your finished image as a standard .bmp file.
Next, open up MS Powerpoint. Create a blank image presentation. Copy the selection of your bitmap image you wish to resize and paste it into MS Powerpoint (note: if your original image is not .bmp then Powerpoint won't recognize it). You can then resize the image in Powerpoint without losing much quality, to a reasonable fraction of the original anyway.
Last of all, make a screenshot of your Powerpoint image; copy & paste it into your preferred paint editor, and resave it again as a .bmp file. You will now have a resized image that does not lose a lot of the original's resolution. If you save the Powerpoint file after resizing within it's own window (in any format), it will lose considerable detail - the screenshot process is key to retaining the detail. This trick will also help smooth out panel lines and other items where pixellation is evident in the original bitmap. Not a cure-all by any means, but it is quick and works well. You can also save the Powerpoint file and resize the images again and again. This has helped speed up progress on my own projects, hopefully it is of help to someone else as well.
(Note: the attached image was saved as a .jpg file and thus loses some additional clarity.)