Bill, Garston station no longer exists! It was knocked down and merged with Allerton(?) to make Liverpool South Parkway.
The airport to Chester is really awkward, there are no direct services and it'd probably take you 2-3 hours trying to bus it and Taxi would be upwards of £60.
The quickest way is to get a taxi to Liverpool South Parkway and the train to Chester (involves one change at Liverpool Central or Lime Street - depends which train you get) which would take about 2 hours in total (including taxi ride).
If you don't want a taxi there's a bus called Airlink 501 which will take you to Liverpool South Parkway (journey time around 10 minutes). They're due every 20mins.
http://www.liverpool...-Parkway-Li.pdf
Total cost of journey around £15 (about half that if you get Airlink 501 - I took the Airlink 500 recently and the service was really good.)
Now, don't say we don't help!!