Sounds fun - it's always very satisfying to repair and.or upgrade an old calssic like that!
Firstly, Bob (above) is right, you can fit QR wheels to your frame, the recesses are a design feature of later bikes and are not needed. No bike in the '80s had this feature.
With the wheels, you've got a choice to make:
Option1: Buy a back wheel with a rear hub that accepts a screw-on freewheel (as Bob says this is not a cassette) which is what your bike would have had when new. You can either buy new or second hand but either way make sure it's alloy, not chromed steel.
Option2: Upgrade to a modern wheel set with freehub that accepts a slide-on cassette with lockring.
Option 1 wins you points for originality and the ease of like-for like replacement. Option 2 gives you a wider choice of gear ratios and the possibility of fitting a 8,9 or 10 speed cassette. I had this choice to make with my every day Raleigh and decided to forego originality and go with option 2.
If you're feeling flush and would like a very flashy pair of Option 2 wheels then I can heartily recommend a set of these
- I've had a set on my Raleigh for the past 3 years and they've been superb - light and extremely strong. Much cheaper wheels are, of course, available!!!
As noted above, your rear dropouts may be too narrowly spaced for modern wheels but it being a steel frame, you may well find that you can just expand the dropouts a tad to get the wheel in - classic steel frames have a bit of give in them.
Once you've decided if you're going with a freewheel or cassette, you can then select a suitable chain and the KMC brand is very good.
In terms of brakes, you've got a pretty big choice - a set of Weinmann 80's alloy ones would look original, or you could go with a set of Weinmann centre-pull calipers with the relevant fittings to your seatbolt and stem. Otherwise all manner of more modern brakes will just bolt on.