I wouldn't get the train - National Express coaches take under an hour from the airport. The coach station at Stoke is also closer to the centre of town.
If the times are inconvenient then you can book a train from airport to Stoke - see nationalrail.co.uk - and book when the cheap tickets become available a maximum of 12 weeks before date of travel. Allow at least two hours between landing and catching the train.
Trains are a doddle check out this page.Just click home to refresh.
It's cheaper and possibly quicker to not go via Manchester Piccadilly.
Manchester Airport to Stoke on Trent is £17.20 single and takes around 1hr15 with a change at Manchester Piccadilly.
On the other hand if you go via Crewe and buy two tickets:
Manchester Airport to Crewe: £9.60 single and takes 35 minutes.
Crewe to Stoke on Trent: £5.80 single and takes 25 minutes
Alanrow suggestion about Advance tickets isn't relevant to this journey as the fare is set by Northern Rail who do not offer Advance tickets.
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