I unfortunately waited too long to reserve tee times but we had a great time and were lucky enough to play some of the courses on our bucket list. We stayed in a little hotel right next to hole 18 on the Old Course in St Andrews, but just up the street a little. It was fine, old, lots of history and quaint, but nothing special. We did not spend much time there. I went with my brother in law so there wasn't any need for anything fancy. My next trip there though we will stay in the Fairmont, Old Course Resort and Spa or the Rusacks just to live the entire experience.
We played the following courses:
Carnoustie - wonderful course. We actually had made a mistake and booked our tee times for September of 2023 and we showed up a year early lol. Gracefully they laughed and accommodated us and got us on about 30 minutes after what we thought our tee times was going to be with caddies and everything.

Kingsbarn - probably my favorite course in Scotland. It would be a must play for me when I return. You will not regret this one.

St Andrews New and Castle courses - We had a tee time at Castle only because we could not get on the Old course as they had a tournament being played there the 4 days we were in St Andrews. We went early and talked to the starter and they were able to get us a tee time the next morning which worked out perfectly for our afternoon scheduled tee time at Carnoustie. I loved both courses.

We were in Scotland just after the Queen died so they essentially shut down the country on the Monday we were there when we had a tee time at Gullane that canceled. So we did a last minute flight to Dublin and got on Portmarnock which was also a fabulous course. We flew back to Edinburgh after our round in Ireland to wrap up the rest of our golf trip.

We then made our way to Gleaneagles to play their 3 courses. Gleaneagles is nice, somewhat fancy, and I did enjoy it there. Not sure I would return though. It was pretty luxurious so we ended up flying our wives who were doing a sisters trip in London over to stay with us for a couple nights at the Gleneagles properties. The courses were more of what we would expect here in America. Not links style, but everything was very well maintained. Definitely worth going once, just not sure I'd spend 3 rounds in 2 days there. I was spent by the final round on the last day of 36. They do have carts there but they charge like $100 per rider or something ridiculous. I guess they try to promote walking which I usually prefer anyway, but these courses were not the easiest to walk like the rest of the Scottish courses.

Come ready to spend a ton of money on golf swag. I love all of my golf gear from there. We had some great Indian Food when we went to Edinburgh at a place called Dishroom. Gleaneagles had some fantastic restaurants at their resort as well. Near St Andrews we got some decent fish and chips at Cromars. We didn't find anything to die for there. We did have a good dinner at Carnoustie though after our round there.

Really wish we could have played Muirfield and North Berwick. Looking at your itinerary I think you'll love it especially if you can get on the Old Course. If slots open I'd suggest Kingsbarn and Carnoustie.

Also we rented a sprinter van. Didn't need it, but was nice to have for clubs and then when the wives joined us. Don't get the manual stick shift, adds an element to driving on the other side of the road when you're also using your opposite hands to shift and steer.