Bicycle Tutor

My old Dawes Galaxy doesn't have a quick release skewer on the rear wheel, is it reasonably easy for me to fit one? or would it be a case of replacing the whole wheel?
I am trying to make it so that I can fix punctures quickly, ideally without having to carry a toolkit around with me.
any help would be appreciated.

Depends on how perfect a replacement you want it to be. First you need to swap the solid axle with a hollow quick release axle, which allows the quick release skewer to go through.
In most cases, the threading and diameter is different between the old and new axle so you cannot use your existing hub cones and must find ones that match the threading on the QR axle. Now unless you can find cones that exactly match the original ones' diameter, height, and bearing race curvature, you'll be met with 1) over-locknut spacing issue as the new cones have a different height than the original ones; and 2) dust-cap coverage issue as the diameter is either too large or small. A close enough match can be found with some looking around, but a perfect replacement is unlikely. Again it goes back to how perfect you want the replacement to be.
You can also swap out the whole hub. But labor/cost wise that's pretty much the same as replacing the whole wheel.