To be perfectly truthful, I doubt you will have enough days to do all that is a' must'.
Chester has to be one day, Liverpool (via Ferry ?) has to be another .
I would suggest a round trip of the Wirral for a couple of days.
Put the champagne on ice before you leave home.
Trip 1) Start off at Shotwick. Read your History
and it won't take very long to see the village, but very interesting.http://www.shotwick.org.uk/village.html
Make your way back to Ness Gardens via Burton village (very pretty old village) and St. Nicholas Church.http://neston.org.uk/the-6-villages/
Ness Gardens http://www.nessgardens.org.uk/plan-your-visit/
Should be lovely in May on a nice day. plenty of places to eat a packed lunch. There is also a cafť ( I think a bit pricey.. last week 2 x coffee and 1 x scone with butter £6 plus !)
From Ness Gardens go either through Neston and Parkgate relate the History
of Lady Hamilton and Nelson( and have an ice cream at Nicholls)http://www.visitparkgate.co.uk/#/history/4552455136
or back up to Chester High Road
Follow the signs for Thornton Hough and stop off for a walk around .http://en.wikipedia.org/wiki/Thornton_Hough
See the Old Smithy, visit the churches (2 of them) and pop into the black and white timber framed Seven Stars Pub
, where you could have a liquid interval of your chioce, or dinner out.
Port Sunlight Village.http://www.portsunlightvillage.com/
Church, Victorian School, Garden centre, Lady Lever . Go to Visitor Centre which gives the History
of the villaage , get a map,(they also do village tours on certain days) and maybe buy a souvenir. There is also tiny tea shop opposite station and near to the Gladstone village theatre .
You could go from there to Brimstage Craft Centre which is only 5 mins away. Have some lunch or afternoon tea in the Country Mouse restaurant, or an ice cream from the ice cream shop there. Book restaurant in advance is safest .http://www.visitwirral.com/food-and-drink/country-mouse-restaurant-p44120
Brimstage Craft centre shops open about 11am I think. So to go there first might hold you up a bit.
If there is time, make your way to Lower Heswall village, where St. Peter's can be found. Back up onto the main Chester High road, and a choice of going to Wirral Country Park at Thurstaston, for a breath of fresh air, or continue along to West Kirby/ Hoylake for a walk along the prom.
Arrive home, sling your shoes off and open the bubbly.
A trip to a castle in North Wales would be great. Conwy is brilliant. I took an American friend there, she was much impressed. There is also a National Trust property Aberconwy House in Conwy.https://www.nationaltrust.org.uk/aberconwy-house/
...and another bottle of bubbly !