Sheldon Brown’s site is your friend. https://www.sheldonbrown.com/k7.html
I put my 1980 Made in Japan, Rayleigh Super Course back together a few years ago. One of my kids liked it enough to bring it bring it to Europe with them but was too broke to bring it back. He got 350 Euros for it so it wasn’t junk.
My bike was 2 by 5. There are a number of different things to do. I kept the original front Suntour BL derailer but the rear wheel and derailer needed to be replaced. The spacing between the rear drop outs on the bike is 126mm. It was a little hard to find a 126mm rear hub and I’m not a wheel builder so I needed a built wheel which made it a little more hard. My bike was steel so I could have stretched the drop outs to 130mm. Your bike might already be 130mm. You should measure what you have. You should determine if you need to replace your hubs or not. Your hub looks OK in your picture so read the section on hubs at Sheldon Browns site and make a judgement. Trueing your wheels is a good idea. If you do want to replace your wheel, I would look for a decent aluminum rim. Given the weight of the rest of the bike, Carbon, if you. can find it, would be a waste.
Look at your freewheel and see if there are notches on it. Contemporary drive trains can be shifted while pushing on the pedals. The old system required letting off during shifts. You can buy new 5, 6,and 7 speed freewheels with the notches. I got a 5 speed from Sunrace.
I replaced my rear dereiler with a medium cage 8 speed shifter. If you stick with a friction shifter, it will work. I had to spend some time to adjust it but it was a lot more sensitive than the old one.
If I still had the bike, I would get friction bar end shifters.
I also got a taller quill stem, wider handle bars, a new front brake, and a new Brooks saddle (which I kept).
This is a lot of work but the bike had value to me. I also like a steel frame and your Raleigh probably has Reynolds 531 steel.
The 126mm rear drop out is the current standard for fixie/single speeds. Personally, the last fixie I rode, had two little wheels in the back. I like mechanical advantage but it is an easier path