Hope you're not going to spend your entire time at the station! (although obviously it's your choice!)
Having just come back from a weekend in Chester, I can recommend a walk around the city walls, very pleasant. That also has some railway interest as the railway cuts through the corner of the walls, isolating a corner tower (those Victorian vandals!). It's now linked with bridges so you can still do the complete 'wall walk'.
Very pleasant riverside walk available too, and don't miss the Grosvenor Park Miniature Railway - I believe they bring out a steam loco on special occasions.
I’d also recommend the short run to Crewe and back, sitting on the right to Crewe and on the left to Chester, as you pass Crewe electric depot, where you can see stored stored 90s in varying states of disrepair.
I haven't calculated the price-per-mile, but £14.80 for an anytime single (cheapest option) for this journey (which I had to make as part of my trip home) seemed high for a 20-minute ride in an elderly DMU - but I don't really understand ticket pricing. I was sitting on the other side so I could catch a glimpse of the APT-P
