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.

I got from my bike shop a QR skewer together with a QR axle which supposedly fits my bikes wheel.
Is there something that should be looked at when replacing the axle?

Go through the steps in http://bicycletutor.com/overhaul-wheel-bearings/ . Also, if this is a rear wheel or front wheel with disc brake, be careful to get the dishing correct (ok, this is also important for a normal front wheel, but much easier to achieve). If both axles have the same thread, this should be easy, just note which washers, nuts, cones go on the axle on which side and in which order. Good luck!

Thanks, that's exactly what I needed (though I don't have any grease or extra bearings available)....

Grease: any that can be used for bearings can be used, get some from a car repair shop.

Bearing balls: some bike stores sell those, they are inexpensive. You can reuse the old bearing balls if they are not pitted and you clean them thoroughly. Cleanliness is the most important thing when overhauling a bearing, any dust, grit, sand that stays in the bearing causes accelerated wear and additional friction. Cleaning bearing balls however is just messy and getting them really clean is tough.

Other things you'll need: rags (lots of them), degreaser, cone wrenches, tweezers.