Trains to/from Edinburgh

You can get a train from Manchester airport to York, I have done it many times. 2 trains per hour.
You can look up train times, and buy tickets, for this and your other rail trips on: http://www.nationalrail.co.uk/
This will automatically re-direct you to the correct train company.

Do not use a travel agent, they will know less than the people on this forum, and may charge you more.

For all your trips (Manchester airport - York, York - Edinburgh and Edinburgh - Oban) you will get discounts if you buy tickets in advance. When you buy advance tickets, the tickets come with a reserved seat included.

For Manchester airport, they do a special deal where you can book a train ticket, and then if your incoming flight is late, you can get a later train at no extra cost. Details here: https://www.tpexpress.co.uk/special-offers/manchester-airport/

We understand that sometimes flights back to sunny Manchester are delayed and you may be concerned about missing the train home you bought your ticket for, don't worry with an Airport Advance you can catch the next available Transpennine Express train up to 3 hours after the departure time stated on the ticket.*

For a good briefing on trains in the UK, see here: https://www.seat61.com/UK-train-travel.htm

I found this train from Manchester Airport to York on October 6.
Our flight is due in from Philadelphia at 8:50 - is 11 am enough time to get through customs, or should we plan a later train? I hate to wait in the airport/train station when York is waiting, but don't want to miss the train either. Also, is this the right train company? And is regular ok or should I change to first class?
Depart Manchester Airport dep. 11:06
Arrive: York, 12:40
Airport Advance Single £18.60
Only valid on booked TransPennine Express services.

TransPennine is the correct rail company.
08:50 to 11:06 should be enough time (2¼ hours). But it isn't going through customs that takes the time, you just walk through the "nothing to declare" door. What takes the time is immigration (passport control), and the risk of your plane being late.

But, if you follow the link I gave you in my last post: https://www.tpexpress.co.uk/special-offers/manchester-airport/
It says:

We understand that sometimes flights back to sunny Manchester are delayed and you may be concerned about missing your train; don't worry, with an Airport Advance you can catch the next available TransPennine Express train up to 3 hours after the departure time stated on the ticket.

So you if your plane is delayed could also get the 12:06, 13:06 or 14:06
There is also a [BUY TICKETS] button on the same webpage.

Don't bother with first class, 2nd class (officially called "standard" is perfectly OK).

A note about the Edinburgh to Oban journey. You need to change at Glasgow.

Edinburgh has the main station (often refered to as Waverley but officially that was dropped decades ago!), plus Haymarket which serves the east end of the city centre. All Glasgow trains call at both so you get on at the most convenient. But Glasgow has two seperate big stations - do not get a train to Glasgow Central!! You need to be at Glasgow Queen Street. And you need to be at ground level, if you get a slow train it may arrive downstairs.

The Oban train will not be a long one, but it may be coupled to a train for Fort William or Mallaig as far as Crianlarich. If there are more than 2 cars make sure you have the right one, otherwise you'll get comfortable and then have to move again!

Thanks - this is very helpful. How many weeks/days before we travel should we book the trains?

Never more than 12 weeks in advance because any cheap Advance deals won't be loaded, in some cases it may come down to 10-11 weeks in advance, notably for weekends. If Advance tickets are available they may be much cheaper than the Flexible Anytime/Off Peak tickets but there may be penalties for alterations later if you change your plan.

We flew into Manchester last year and were on a train within an hour or so tops. Very simple. Station is right there. Walked up and bought tickets to Lincoln. Easy peasy. We also trained from York to Edinburgh - about a two hour trip. We were on Virgin East (?) line which went right along the coast and was very scenic. We did not buy our train tickets until we were in the UK and were still able to score quite a few advance fairs. If you buy in advance, you must have the same credit/debit card that you paid with in order to collect tickets at the station. My son found this out the hard way this summer.

A tour guide for Edinburgh might be helpful to you. We used Bill Hill and was thoroughly satisfied with him. See: https://www.edinburghtourguide.com/