Yes, choosing an appropriate format is important.
Photos: Use JPG with a moderate compression level.
Screenshots: Use PNG (if it doesn't contain a photo). In this case PNG compresses more efficiently than JPG and doesn't produce artifacts.
Graphics with simple structures: Create them in SVG format. It will scale up without loss in quality.
Don't use an original image produced by your camera or mobile, these are too large.
Do you host it on a faster dedicated place?
I never did so.
Yes, BG images covering the complete browser window are popular but a large size is required causing a long loading time.
A workaround might be to load the the BG image only when loading of the rest of the page has been completed.