It may be worth changing your route slightly as you may be surprised how cheap flying with a budget airline like Ryanair
can be from Dublin to Liverpool..
But if you dont want to change your flight, then for the route youve chosen, for the short distances i dont think its necessary to book your train tickets in advance. For the journey from Chester to London though the price sometimes triples if you leave it to the last minute, this maybe worth looking into see www.thetrainline.com
Chesters a beautiful place and defo worth a wonder around,
You can have an option to pick up tickets from the train station, I know you can do this at Manchester train station not 100% sure from chester tho!
hope this has helped!