This doesn't make sense. I don't believe an image is going to mysteriously get larger as a result of having been sent as an email attachment. At least, I've never seen that happen, and I've often emailed photos. So, I'm inclined to think some basic facts may be missing here.
Anyway, I would use the JPEG format for emailing. A JPEG image file is likely to be much smaller that a .TIFF file, based on what I've seen working with Paint. I saved a small image in both formats, using Paint. The JPEG image was 4,355 bytes, while the TIFF image was 40,244 bytes. Be careful to keep your original image; don't overwrite it with the downsized JPEG.
Since you've not had previous experience sending attachments, I suggest you experiment with a few simple text files (you can use Notepad for this), and some small images to see whether they go through.