You would be wise to camp out at least four days at the Mallards Hotel in Gullane - the rooms are comfortable with a small Pub downstairs. A combo bar restaurant called The Old Clubhouse is a three minute walk up the street and the food is damned good. There is a more formal restaurant attached to a hotel on the main road. I think it is called "The Golf Inn," but my memory is a bit sketchy these days.
Mallards is right across from all three Gullane tracks and Muirfield is just down the way if you can swing a tee time. Archerfield is right next to Muirfield and North Berwick is 10 minutes away. That will keep you busy for a few days. Glen Golf Course (North Berwick) looked pretty neat - a quirky shorty right on the water. Make sure you play North Berwick at least twice - it is a magical place much like Prestwick or Lahinch.
I've got to echo another poster about a Sunday stroll on TOC. Bring a putter and a couple balls, nobody will give you a hard time. The Ladies Putting Course after dinner with the homies is mandatory; bring a flask of Single Malt.
When you get over to the West Coast, Troon is last priority. It has four interesting holes and 14 ducks. Plus, the club is run by a bunch of assholes who make no secret of their disdain for Yanks. You might want to play The Portland Course, it is a lot of fun and terribly underrated. Whitten is not a fan of Turnberry, but we liked it well enough to stay over and play it twice. We did not play the Kintyre Course, but I've been told it is worth a go.
My friend told me recently that we screwed up not arranging a game at Western Gailes - I'm sure somebody on this board has a detailed opinion. One other "name" place you might consider is Gleneagles. Our group split up and - despite my childhood buddy having won the Scottish Open on the King's - I chose The Queen's Course on the recommendation of a close friend who knows my taste. It was wonderful . . . . The hotel is ungodly expensive, but we split the rooms and to be honest, staying a couple nights in a cha-cha 5-Star was nice after bunking out in a series of marginal B&B's. Scottish B&B's generally suck in comparison to Irish and English ones. Sorry.
As for the Centenary Course, I wouldn't walk across the street while in Scotland to play a Nicklaus Course if Nicole Kidman was going to caddy for me naked. Depending on how much time you have to go north, Goodale is a member at Brora and Dornoch - Nairn is supposed to be terrific, too.
All this stated, you've made a life-changing mistake if you skip Cruden Bay. We flew into Heathrow and as it took three hours to clear customs, missed our connector to Aberdeen to meet the rest of the group. By the time we got it all sorted out and finally arrived, The Redhead and I had not slept in 34 hours. I jammed to Cruden Bay and the plan was for Her Redness to pass out in the little hotel while I met everybody for their 2nd round.
We pulled into the parking lot and I could scarcely believe my eyes. It is every bit as spectacular a sight as the first view of Portrush from the saddle road. Redhead opened her eyes, took one look, put on shoes and grabbed her camera. I swear to God the colors at Cruden Bay seem brighter . . . . it is worth crossing an ocean just to go round and round that course.
Depending on how you get to the U.K., playing golf around London is not to be missed. 36 holes on Sunningdale Old & New was incredibly memorable. Heathland golf is a completely unique experience; wandering around a course with the heather in bloom will lower your blood pressure by 50 points. Start by flying into Gatwick, Heathrow is nearly as big a mess as JFK.