I meant easiest in the sense of a foreign tourist, being unfamiliar with the system, boarding a wrong train or alighting at the wrong place and then being accused of being a common criminal for having made an honest mistake.
Buying a ticket to Liverpool and then boarding a train to Liverpool and riding it to the end. And then buying a ticket to Chester and boarding a train to Chester and riding it to the end is pretty safe and not confusing, even if it takes a little longer.
It's not like you can ask a friendly station porter for information any more. Totally different for railway experts of course.
In that case the best bet is to go by one of the following routes and buy the appropriate ticket at the ticket office ("via Crewe" or "via Manchester").
For most of the day there are two services every hour:
xx10 from the Airport to Crewe arr xx45, dep xx23 gets you to Chester at xx46 (1h36m).
It usually arrives in platform 1 at Crewe and you have a 38 minute connection to walk across the bridge (stairs/lift) to platform 9.
There is also a chance of catching the earlier xx49 Virgin service to Chester and saving half an hour, but not guaranteed.
xx25 from the Airport to Blackpool gets you to Chester at xx53 (1h24m), with a same-platform 5-minute change at Manchester Piccadilly (arr xx48, dep xx53) no need to move, just get on the following ATW Llandudno train).
There's a through ATW train from the Airport at 0533 (Chester arr 0649) but that probably won't be any use.
I'd ignore alternative routes, even if they are a bit cheaper.
They involve stopping trains and (usually) poor-quality interiors.
We don't want a foreigner's first experience of British trains to be a Pacer!
The Northwich route also involves passing close by the airport about an hour after you left it heading for Manchester!
And don't get the tram, it is very slow
Another factor for a newbie is how to reach the Airport rail station ("The Station").
Access from Terminal 2 is fairly easy, but I find access from T1/T3 indirect, confusing and poorly signed.
Even when you reach The Station it's not exactly intuitive how to find the ticket office and the trains.