By plane
Manchester Airport is located just north of Cheshire and serves a large number of international and domestic destinations. Liverpool John Lennon is similarly located nearby and has mainly budget airline connections to European destinations. It will not be possible to get to most places in Cheshire direct from here, rather you must go to the main train stations in that city and use that.
By Rail
Crewe station and Warrington Bank Quay are on the West Coast Mainline, which gives them good connections to London, Birmingham, Glasgow, Edinburgh and a variety of other destinations. Chester has more infrequent services to and from London and the Midlands, and regular services to and from North Wales, Manchester and Liverpool. These last three are largely commuter lines, so do not expect major luxury, though they pass through enjoyable country scenery.