Malcolm,
I have a very busy summer ahead but I hope we can meet up for a pint or two at the very least. You will be living about 5 minutes from my house.
Ok. Get a car!
Play the Braids here in Edinburgh, about 4 minute drive from your flat. Municipal, very cheap, lots of fun. Bruntsfield and Royal Burgess are nice historic clubs and not too pricey. You can get to all of these on the bus.
In East Lothian, play Musselburgh Old for the experience but then focus on Dunbar (40mins), Kilspindie (30 mins), Craigielaw (30mins) The Glen (40 mins) in that order for great value golf. Goswick (70mins, further afield but exceptional value for a round or the day.
If you can get in at Gullane then 100% jump at that opportunity. No. 1 and No. 2 have some really good holes and you will not be disappointed. On busy days you can play No. 3. Short with enough interest on the greens to keep you interested. Since you will effectively be a member of 3 courses the value is off the charts. You will always be able to get out on one of the courses. It will no doubt be one of the best membership decisions you will make in terms of value and enjoyment. Even if you can't get in as a member they have some great deals and I would rank all courses as good as the ones above. No. 1 and No. 2 being a nose ahead.
Do the day ticket at North Berwick and you will love every minute. Wait for great weather but its usually good down there. Take the bus and train so you can have a few drinks. Luffness New is in that price bracket too. I'm a big fan of the greens there but NB is way ahead of all the courses mentioned.
Clearly Muirfield is worth it if you want to treat yourself. Have a look at their website but I fear they may be fully booked for the whole year. Expense to me is relative. Yes, its expensive compared to the other Scottish courses but its a better course than Pebble Beach and much cheaper....and a quicker round.
There are better informed guys for Fife. My Fife favourites outside of St.Andrews/Kingsbarns are Elie, Crail Balcomie and Scotscraig. I can't remember much about Lundin or Leven other than I enjoyed them. East Lothian is probably 30 minutes closer than any of the Fife courses from where you are staying.
Hope that helps.
Simon