Train from Edinburgh to Inverness

I would not wait. I don't know about Saturday mornings specifically, but trains can sell out. Even if they don't, you may find the ticket cost a great deal higher as your travel date approaches. You can check that out now on www.nationalrail.co.uk. Price out the ticket for tommorow, for his coming Saturday, and for your actual travel date. I think you'll see what happens to prices in the UK when you buy last-minute tickets.

Trains do not sell out.

On the day, you can get a walk-up Anytime ticket for £50. Take any train tickets, always available.

However, you can buy cheaper priced Advance Tickets from about £18. These are on sale now and do sell out. You must travel on the train you book.

https://www.scotrail.co.uk/

I'm afraid that there is no chance of you getting there that morning.

If you touch down and everything goes your way you won't be at Edinburgh Waverley station before 9 or 9:30. The train trip from Edinburgh Waverley is over 3 and a half hours.

The trains that day are:

6:38 - nope

8:34 - nope

9:38 with a change at Perth, arrives Inverness 13:25 (maybe you might make that one?)

10:35 if you plane is on time you may well make that one arrives Inverness 14:11

11:34 with the change at Perth arrives Inverness 15:22

There are "Advance" tickets available for all these trains today, as low as £18.30 pp for the direct trains, from £24.80 for the ones with the change in Perth. HOWEVER - these are only valid on the exact train purchased. If your flight is early or late, or if you have long queues at the Border Force checks or it takes a long time to collect luggage you cannot change the train time. You will buying a new ticket, and if you are caught on the train with the wrong ticket expect a substantial fine.

The full walk-up fare ("Anytime") is £58.60, an Off Peak Return (round trip) is £75.00.

So you need to decide how brave you are - if money or convenience is more important to you. Can you afford to both lose the Advance tickets you bought and buy full priced ones to board the train you can get, or do you prefer to be conservative and allow plenty of time that you might have to fill in Edinburgh while you wait for the appointed time of the cheap ticket?

In any event, you see that you won't get there before the afternoon, probably mid-afternoon.

If you touch down and everything goes your way you won't be at
Edinburgh Waverley station before 9 or 9:30. The train trip from
Edinburgh Waverley is over 3 and a half hours.

If you arrive at Edinburgh Airport it would be a bit of a detour to go to Waverley. It would in my opinion be a better idea to catch the train at Edinburgh Gateway or maybe Inverkeithing.

Nigel, I might not have been clear, I'm not that concerned about getting to Inverness in the morning. We just want to be on the way sometime in the morning.

I don't mind paying a little more if it means I don't have to be rushing to catch a specific train. We're on vacation and we make sure our vacations are well funded. We are just trying to get there with as little problem as possible.

We don't need to go to Waverly Station? If we can get our tickets at the airport and leave from there, that would be great. We are working our way back to Edinburgh eventually and will be staying for a few days there before we leave.

There is a tram that goes from the airport to Central Edinburgh. It stops a couple of blocks from Edinburgh Waverly. However, it stops at Edinburgh Gateway station. The trains from Edinburgh to Inverness stop there. It's much smaller and easier to manage than Waverly

All trains to Inverness do not stop at Edinburgh Gateway
I haven't checked all trains but it appears only the ones that involve a change at Perth stop there
Inverkiething is the other side of the Firth of Forth!

All trains to Inverness do not stop at Edinburgh Gateway

I checked the trains to Inverness on 23 April and they all pretty much stop at Edinburgh Gateway--both those that change at Perth and those that don't. There may be one or two that don't but the ones, in the morning, that the OP wants do stop at Edinburgh Gateway.

just a couple of issues getting on at a small station like Gateway -

first although the travel time is something like 9 minutes shorter, that gives the main group of passengers, those who joined at the origination station of Waverley, and those who joined at Haymarket, a chance to find the best seats and combinations and have settled in by the time our OP joins the train and settles for the leftovers. I find it best if I am about to start a 3 and a half hour journey to join the train some time before departure and choose seats which are in good condition, have time to adjust the storage of my luggage, sit where the window lines up properly (and is clean and free of graffiti), and spread out before everybody else piles in.

second, the train stops for less than a minute at Gateway, just long enough for a couple of folks to get off, the new ones get on, and as soon as the platform is clear off goes the train. Luggage adjustment when the train is moving is never as easy as when it is stopped, nor is shopping for seats.

Other than that it does save a few minutes.

I expect that Waverley has much more choice for on-train refreshment provision (stores to you and me) than Gateway.