There are many resources for cottages in both countries - from agencies, to private owners, to the National Trust, to even time share properties.. But before any specific recommendations - what is your budget?

Also - while 2 weeks in two locations would be great -- I'd also seriously consider 4 one-week rentals. You can see a lot from any one base, but you could see that much more from two different, well located bases in each country.

"within a short distance of some type of public transportation, so I can use the trains to go about; "

...simply don't exist in western Ireland or the Highlands. Neither, incidentally, do non-affordable cottages, unless you're on an impossibly frugal budget.

It's just possible - but involves limiting what you see immensely - to do some touring by bus, ferry, walking and cycling in rural NW Scotland (though I swear I'll never do it again). Whenever I've tried it in rural Ireland, it meant in practice lots of reading and writing in a little more than OK house, with a couple of walks and one or two messy excusrions based on the once a day local bus.

In neither case are trains of any use at all, except to get to a couple of Scottish towns (like Mallaig) from the outside world.