Thank you for all your replies! I managed to get round to rechecking what I actually have on my bike.

1. My current bike is a three speed Raleigh Caprice (and my previous had been a three speed Peugeot city bike) which I mentioned to give you an idea of the simple set-up I am used to. The thought of too many gears and two gear shifters is quite daunting to me! The bike I have just bought to fix up and I am asking about is a very old Reynolds Falcon. Photos below.

2. The ring on the crankset looks like it is not a single piece with the crank arm, but I do not have the right tool at the moment to try and take it apart. So I guess I could see about just getting a ring, or a double ring. I am not quite sure about the rest of the stuff mentioned, I am only now learning about all of this stuff!

3. I completely miscounted the amount of sprockets I have at the back! It turns out they are only five. The biggest is 28 and the smallest 14. Due to the age of the bike I assume its a freewheel. I don't quite understand how you can tell from looking at it.

Now that I realise I have 5 gears at the back, I am re-thinking things. Should I just get a classic double crankset for the front, even though that would mean adding a second gear shifter and getting hold of an vintage old-school style one might be very difficult? Or should I go for a smaller (so many proposals!) single crankset in front and add some more sprockets at the back. I am assuming that freewheel means you can add single sprockets as needed since they are not a joined block.