Bluey, have a look at the settings in Corel. JPEG files are always compressed when they are saved, and playing with these compression settings will probably get you the result you want. I haven't used Corel in ages, but somewhere in the software, either in the 'settings', 'options' or 'preferences' (or whatever they're called) dialog, or in the 'export' dialog (might be under a 'more options' button) there will be a 'JPEG compression' or 'JPEG Quality' setting. On a scale of 1-100% I find that somewhere between 45 and 75% will get you a decent tradeoff of quality against filesize. Something else that might help is to reduce the image size before saving, unless you need a large size image many people will not be able to display an image over 1000 pixels wide at full size setting anyway.
Winzip won't help you since a JPEG file is already compressed. The only usefullness of Winzip in situations like this is to bundle several images in one file. The filesize will not change by more than 2% though!
As for image types, JPEG is the way to go for e-mailing purposes. GIF is better suited to logos and digital images which use large areas of a single colour. TIFF is a file format that can store a lot of information (layers, pen paths etc), and is therefore not suitable for e-mailing since TIFF files can get quite large.