Write the same kind of cover letter when emailing work as when mailing, with your reasons for submitting to them (provided you have sincere reasons particular to that publication, of course) and a short bio. As with snail mail submissions, be careful not to send the same story to the same journal more than once. Most journals are only published a few times a year: refrain from submitting to a publication more than once per reading period (usually spring and fall). With the New Yorker, this doesn't quite apply. Nonetheless, I'd still keep it to every other month on the outside.
For more publications -- if you've been through the lists here -- you might look at a copy of Novel and Short Story Writers Market for others that take email submissions (most libraries have a copy). And look into journals and publications in your area, even at your college, where you might feel that the postage is worth it. But don't feel that you have to be submitting at this stage, or any. It's OK to focus your energies on the craft of writing and save the publishing grind for later.