Match you and raise you one m'learned friend as there is not a single Scottish ground, either league or non-league, extant or extinct that I have ever visited.
I have often thought I should have this Caledonian experience but not much takes me north of the Border apart from weddings and a single visit to Murrayfield.
I wonder which club I should visit? A dear late friend's late father used to play for Kilmarnock before the war and so perhaps that should be my choice.
Had you chosen to go today it could have been an interesting report - 3 Kilmarnock players dismissed and the Kilmarnock manager calling it the worst refereeing performance he had ever seen in his career.
You might also want to avoid planning a Kilmarnock - Rangers fixture if a recent headline in "The Scotsman" is to be believed; "‘Rangers might say they won’t play at Kilmarnock any more’ - Ex-SFA chief says Gers could boycott Rugby Park".
https://www.scotsman.com/rangers-2-17785/in-full/rangers-might-say-they-won-t-play-at-kilmarnock-any-more-ex-sfa-chief-says-gers-could-boycott-rugby-park-1-4908865Other than that Rugby Park is not a bad ground, I'd guess about 25-30 years old. The Park Conference Centre and Hotel in the car park of the ground is quite reasonable with good food in the "Blues" restaurant. Kilmarnock town centre has, like Ayr, not a great deal to commend it - though there are good places to visit in Ayrshire - Prince Charles favourite Dumfries House near Cumnock (You could also try out your Parliamo Auchinleck at Beechwood Park), Culzean Castle (a favourite of President Eisenhower) and Burns Cottage and the Auld Kirk Alloway of Tam O Shanter fame are all worth visiting.