Thinking of a Scottish golf and tour experience for May 27. Suggestions?

First thought- flying into EDI is cheaper than Heathrow. Heathrow involves high airport fees. I know you'll probably have a LHR connection, but EDI as a final destination works better.
We are golfers, and I have never seen an RS golfers' tour. Remember that RS hotels may have steps, no lifts, and the rooms can be small. Also, sometimes the RS bus can't drive into the city center, so some walking with suitcases can be involved.
Lugging clubs is a pain. It helps if you have one of those very light, soft-side, gliding travel golf bags. The heavy coffin-like club carriers are a disaster. We are speaking from experience. My husband took clubs a few years back, and he constantly needed help from my daughter and I, when using public transportation. ( This was not a RS tour.) I did not take my clubs.
How many sets of clubs might be going?
Rental cars are small in Europe. Clubs and suitcases take up a lot of room in those small cars.
When my husband did a boys (four men) golf trip in Scotland precovid, he just hired a car service for all the transportation.
Green fees have gone up significantly in the past few years- be prepared for sticker shock.
But- if it's all worth it to you, have a great trip, Fore!

There are direct flights to Edinburgh from Chicago and Toronto, as well.
As a starting point, I recommend identifying the courses you want to play. From there, you can develop an itinerary and determine accessibility by public transportation. For example, St. Andrews doesn't have a train station. It is served by one in Leuchars, with connection by bus into St. Andrews. It's not far, so a taxi probably wouldn't break you.
Re: rental clubs, I would read the reviews of the various courses to determine what kind of quality you can expect. An option might be to take just your driver and putter. Traveling by train with a big, heavy golf bag sounds like a hassle, but if you're not an enthusiastic "wrong side" driver, it's best to stay off the roads.

If you are planning to travel by public transport, rent your clubs, Why drag your own around for a week or 2 if you're only playing a couple times?

I would forget Rick's fast paced itineraries and create your own based on your personal interests in addition to golf-history, castles and ruins, whisky, Scottish ancestry, ancient Cairns and Stones, Hairy Coos, hiking ...

You will find golf courses in every part of Scotland, many local courses with reasonable fees and sometimes fantastic views that will welcome you in addition to the famous courses.

When planning your itinerary, keep in mind that you can walk the Old Course at St. Andrews on Sundays.

If you can get flights to Edinburgh then do it!

Taking clubs on the mainline train from London as an individual isn't too bad (as a group is a nightmare for everyone else!!) but you have to reach the train first and that will be hard work with clubs and the rest of your luggage. Without clubs, much easier - for instance there are 2 or 3 fast trains per hour from London Kings Cross to Edinburgh, around 10:00 (probably too early from your flight) 14:00 and 16:00 they go forward to Aberdeen with a stop at Leuchars. The one at 12:00 goes forward to Inverness including a stop at Gleneagles.

Edinburgh airport isn't as well connected by train as the Scottish Government would like us to think but the places you are likely to be heading for are close enough to make a private transfer be viable, especially with clubs, and you can use the trains and express bus network for your sight seeing. As far north as Perth/Dundee you can consider public transport to be intense by US standards and even further into the Highland or along the coast to Aberdeen it's still frequent.

I'll make one specific recommendation. Dunbar, East Lothian. It's about a 30-minute train ride from Edinburgh, and it's very nice town with a links golf course and a ruined castle in its working harbor. I don't golf, so I can't comment on the quality of the course, but you won't regret visiting Dunbar. It's not on most tourists' radar. Because it's so close, you could do it as a day trip from Edinburgh or nearby North Berwick (though the train doesn't link these two; bus only).

also bear in mind that the first and last Mondays in May are public holidays and trains from London will be even busier than usual from Thursday afternoon onwards. Accomodation may be fully booked well early those weekends

Count me as a vote against traveling with golf clubs. Did it through Scotland and Spain and never again. It is true when considering traveling with luggage, “the less and lighter the better.” If I really need my own clubs at a distant location, I look into shipping them. In most cases you are not going to score that well and the experience is what counts.

As to courses, naturally you want to play one of the St. Andrews courses and consider renting a car out of Edinburgh and playing some of the nearby great East Lothian courses east of Edinburgh like Muirfield or North Berwick. If those are too pricy or hard to reserve, play one of the 3 courses at the Gullane Golf course. All provide the links feel and great views and Gullane #1 is a really good course that is well regarded. Naturally make advanced reservations unless you just try and arrive and get a booking as I did at St. Andrews. All courses there are fun and the rental clubs are good. I took a used pair of shoes and gave them to local playing partners after my last round to free up some luggage space and weight.

@Jim
" In most cases, you're not going to score that well"....
If those aren't wise words!

Scotland Golf Tours offer packages but will also mdesign an itinerary specially for you.

Golf around Scotland offers a variety of golf tours around Scotland. Some are based in the same place , others do travel around (and car would be probably be needed to do this...)

Your Golf travel has ideas for one night stays offering golf.

Not a golfer, but I'll share an anecdote that might help you decide. When we were waiting to board our flight in Edinburgh to return to the US in May, we had a conversation with a fellow traveler. He had come to Scotland for an 11-day golfing trip. His clubs caught up to him on the 9th day! I'd say rent, especially if you're going to be staying in multiple locations, and if you'll be trying to use public transportation.