According to Facebook guidelines, high-resolution images can be as large as 2,048 pixels (wide). This particular dimension will produce the best visual result—even if Facebook's automatic compression algorithms, which are added to every upload, are heavier (more damaging) than most people might be comfortable with. Unfortunately, there's nothing you can do about the compression level. If the images don't appear as good as you hoped, try re-editing them and upload them again—this only takes a few minutes. Facebook is the world's largest repository of images, so, with something like 300 million images uploaded every day, I'm surprised that its compression algorithms aren't even more aggressive.
One typical method of reducing the resolution of an image that is destined for posting on a personal website or social media would be to use the Image Size tool (Image | Resize | Image Size). This is fine for one or two images but tedious...
