Steps for achieving reasonable image (figure) sizes in Adobe Illustrator (and probably other programs too)
Set the canvas size to what you want the final image size to be. For print publications, this is usually one column (~86 mm) or 2 columns (~180 mm) wide. (This ensures you don't save a larger image than necessary, and also helps you to plan how things will look on the printed page...i.e., 11 point font will be printed as, well, something close to 11 point font).
Export > as PNG. For actual print, you want 300 ppi. For screen display (or to embed in a manuscript), you can probably get away with 72 ppi.
For line art, saving figures as EPS files or PDFs will almost always provide a superior result, and frequently be a smaller file size. So, you can try embedding these, too.
(The exception is that if your EPS file has a lot of raster images in it, particularly if you have resized them to fit. In this case, you will want to either resize the images before adding to your Illustrator file, or simply save it as a PNG (which will render the final size, and it won't matter that your Illustrator file is keeping track of the large original size.)
If your PNG image is still too big
...you can try saving to a JPG. JPG is a compressed format and you will lose some image quality, but it still may be good enough. You can tweak this by starting with "maximum" quality and gradually reducing the quality until you get a file size you are happy with.
