Liverpool or Manchester?

We stayed in Liverpool when we attended The Open in 2023. It was convenient as they were playing in Hoylake. The day we did not attend the tournament, we enjoyed exploring Liverpool. We walked to the cathedral and then down to the Royal Albert dock. We also rode the ferry as we enjoy being on the water. The Ropewalks area was also nice. We did not spend any time in Manchester, but I would recommend Liverpool.

I've been to all three cities. Each has what you seem to seek.

My advice is to spend some time researching all three cities, what they have to offer, and then decide which one seems more interesting to you. Each city's history is different.

Otherwise all you will get is favorites from others whose interest might be different than yours.

There is a train twice an hour direct from Southport to Manchester, taking around 75 minutes.

One of those trains every hour goes to Manchester Victoria, the other to Deansgate and Oxford Road (terminating there).

If you go into Liverpool or even to Chester then the most economical way to do it by train is to buy an All Areas Merseyrail Daysaver ticket for £6.40 (not valid on weekdays between 0631 and 0929)- https://www.merseyrail.org/tickets-passes/daily-travel/day-saver/

Another place you can easily reach by train (on the way to Chester) which meets your criteria is Port Sunlight (on the Wirral)- also with the Daysaver Ticket- https://portsunlightvillage.com/

this really depends on how much time you want to spend on the train. But Liverpool seems to be the best idea at only 40 minutes, those train also call athe local stations south of Southport so check exactly where the accomodation is. Chester would add another 50-60 minutes to that plus a transfer at Liverpool Central (at Moorfields on the way back).

Or another way to decide, wait for an accurate regional weather forecast on the day and go to whichever of the three has least chance of rain!

I’ve spent time in both, as well as Chester. You’ve been to Chester so you know that one. Looks like Liverpool is closer than Manchester or Chester to Stockport. Some direct comparisons of attractions given your interest in history and museums:
Social history museums: Museum of Liverpool or People’s History Museum in Manchester? Both worthwhile but Liverpool one is better IMO.
Military history: Imperial War Museum North in Manchester is very good, but Western Approaches Museum (kind of a naval version of London’s Cabinet War Rooms) in Liverpool is unique. IWM North isn’t walking distance of central Manchester but an easy tram ride.
Transport and technology: Manchester Museum of Science and Industry was mostly under renovation when I visited and was not impressive. Liverpool Maritime Museum very good but is currently closed for renovation.
Art galleries: Walker in Liverpool and Manchester Art Gallery both ok but not must sees. London far superior for that.
Rylands Library and Manchester Cathedral are worthwhile and very near each other. Didn’t see any churches in Liverpool.
Everything I’ve mentioned is easily walkable from the main train station in either city except IWM North.
Liverpool has the Beatles sites and Manchester a popular museum of English football but neither of these interested me.
Central areas of both cities are lively and pleasant to walk, with plenty of pubs, restaurants, shops etc.

Didn’t see any churches in Liverpool.

There are basically 3 Cathedrals- the Anglican one, the Roman Catholic one (affectionately known as Paddy's wigwam) and the undercroft to the Roman Catholic Cathedral- one of the great Churches that was never built beyond the undercroft. It would have been the 2nd biggest Church in the world. Totally different to the Cathedral which was eventually constructed, and an amazing space. Yes you have to pay a few £ to go down there, but worth it.

There is also St Luke's bombed out church- a legacy of the terrible WW2 bombing Liverpool suffered- https://www.slboc.com/

Instead do Chester

I think Frank II is right in that you should go with your instincts as to what you want to see. They all have their advantages and disadvantages. Me.crewe also makes a good point in that it would take you a bit longer to get to Chester and you'd have to change trains, whereas Liverpool would be easier. But really think about what you feel like seeing.

In fact, one option you might consider is to just play it by ear. None of those trains need to be booked in advance. Wait until the day before or even the morning of and see what you feel like doing. I find that works for me quite a bit when I am on a trip.

I should first of all stick up for the town of my birth, Southport. Sadly, it is a long way from its 19th century heyday as a high class Victorian resort although Lord Street still retains plenty of it's former elegance with verandas along a lot of it's length. It's still not clear how true the tale of Lord Street inspiring a young Napoleon III when he visited whilst in exile in the 1840s to instruct Haussman to build Paris' famous boulevards. It's definitely worth a look although you may well be able to fit that in on one of the days you will be at the Open.

For your day out you won't go wrong with any of the 3, although Liverpool will definitely be the quickest to get to. Chester really is a bit awkward in comparison.

If the day is good you might want to divert on the way into or back from Liverpool to see Anthony Gormley's Another Place on Crosby Beach -https://www.visitliverpool.com/listing/another-place-by-antony-gormley-at-crosby-beach/16098101/ It's about a 15 minute walk to the beach from Blundellsands & Crosby station. They are best seen at low tide, as many of the statues disappear under water at high tide.