In Which Order?

IMO, your transatlantic flights are always the starting point. You're likely to find direct flights only to/from Dublin and London, so find the ones that work best for you. You don't have to fly open jaw. You can book separate one-way tickets on different airlines if that suits you.
You can travel between Dublin and Galway by train (to/from the city) or bus (to/from Dublin Airport). If you don't want to backtrack through Dublin, you could use Shannon airport to/from either Edinburgh or London.
You can travel between Dublin and London or Edinburgh by plane or by a combination of ferry and train. Here is some info:
https://www.seat61.com/train-and-ferry-to-dublin.htm
I'll leave it to Stuart (isn31c) to get you to Grassington from either Edinburgh or London, but that seems like the order.
If flying into Dublin: Dublin, Galway, Edinburgh, Grassington, London; the reverse if flying into London.

All else being equal, I'd start in London and finish in Dublin. Dublin has US preclearance so you'd go through US immigration and customs there and then land in the US as a domestic flight. It'll save you from the queues in the US.

@G3rryCee's answer is interesting as my first thought was to fly United to Shannon connecting at O'Hare.
Shannon to Galway use the hourly #51 bus.

However it 'should' be cheaper to arrive into London and leave from Dublin or Shannon as that way you avoid paying UK departure tax, and do US pre-clearance before boarding in Ireland.

Either way Ryanair fly from Shannon to Edinburgh.

From Edinburgh to Grassington my far preferred route by every metric would be train to Carlisle, change for the train to Skipton via the Settle/Carlisle line. Then the more or less hourly bus #72 from outside the station to Grassington.

However you could take the hourly train direct from Edinburgh to Leeds, connecting there for Skipton as below.

Grassington to London there are in essence four routes in my personal order of preference-
a- Bus back to Skipton, train 6x a day to Lancaster for the hourly train to London Euston
b- Bus back to Skipton Bus Station, hourly bus #280 for the two hour scenic ride to Preston for the same hourly train to London Euston
c- [on a Monday, Wednesday or Friday only]- bus #74 to Ilkley for the half hourly train to Leeds, for a connecting half hourly train to London Kings Cross
d- Bus back to Skipton, half hourly train to Leeds, for a connecting half hourly train to London Kings Cross. There is a once a day Skipton to London Kings Cross train. But it leaves too early/arrives too late for a bus to Grassington, so you would have to use a donkey (sorry, taxi) for that leg.

If routeing via Carlisle or Lancaster you would book the fare in two halves- Edinburgh to Carlisle, Carlisle to Skipton; Skipton to Lancaster, Lancaster to London.
Routeing via Leeds book as Skipton to Leeds, Leeds to London.

As this is a retirement present book first class on the Lancaster or Leeds to London trains, and enjoy the included meal, alcohol and more room.
Leeds, Kings Cross and Euston also have pre and post journey first class lounges.

Personally I’d base the flight on the best cost option and as I fly from LAX has to be non stop.

We just returned from a similar trip but focused on small towns. Landed in London, trains through GB, flew from Edinburgh to Dublin, train to Tralee and rented a car only for driving around Dingle, train back to Dublin and flew home. The pre-clearance in Dublin saved so much time upon arrival as we just got off plane at a regular gate without need for customs or immigration. The trains in GB were easy to book and you can get a two together Railcard to get 33% off each ticket. You can get the same discount for seniors if you are both over 65.

You can get the same discount for seniors if you are both over 65.

Over 60

We did something similar in September and October 2023. Flew from Newark to Edinburgh and spent six days and used train and buses for transport and two different day trips to St..Andrews and Loch Ness. Then train to London for eight days again all public transport metro, buses, train and taxi. Perfect trip. We then went to Paris by Eurostar and home.from there. On a different trip we went to London and flew to Dublin and home.from.there. All.of this was very easy with a little bit of planning. We stayed very close.to Waterloo station in London as we knew we.would be talking numerous train trips so.this worked.out well. Someone mentioned buying one way tickets instead of multi city and in the past we checked that option but it was very expensive to do the air fare that way. Just check the prices. We did use a taxi a.lot more.than in the past for convenience since I had broken my back and needed to.conserve my energy at times and that also.worked.out well. Best of luck and enjoy.

Thank you all…this forum rocks! Special thanks to isn31c for taking the time for such awesome detail in your response. I feel like we’re in very good hands for that leg of the trip! And G3rry Cee thanks for the article. I will take some time to comb through that!