ISO info on Liverpool

Presumably you are on the Isle of Man Steam Packet Ferry from Dublin via Douglas, as there is no other passenger ferry from Dublin to Liverpool now, and the Belfast ferry berths at Birkenhead on the other side of the river.
The Steam Packet does berth close to the Museum of Liverpool.

I assume you meant one or other of the two Liverpool Cathedrals (Anglican and Roman Catholic) as Durham Cathedral is on the other side of the country, on the East Coast. A direct train from Liverpool, but a 3 hour ride each way.

The highlight of my visit to Liverpool was the National Trust tour of the Beatles Childhood homes. It was an excellent glimpse into everyday life in the 1950s, as well as an opportunity to learn about the early days of the Beatles. You absolutely have to book this in advance.
https://www.nationaltrust.org.uk/visit/liverpool-lancashire/the-beatles-childhood-homes

I stayed at the Resident Liverpool and was happy with my stay. It is a short walk from Bold St, a street filled with small restaurants.

I spent a few days in Liverpool in 2023 and really liked it, despite having no interest in a Beatles tour. The Museum of Liverpool is good and has a Beatles exhibit, although if you’re into them you will want a tour. There’s a Premier Inn just across the street from Lime Street station which is fine and convenient. Some decent restaurants in that area and it’s just a 20 minute walk from the Albert Dock area. Unfortunately the Maritime Museum is closed for renovation I hear, but there’s plenty else. If any interest in WW II see the Western Approaches Museum, kind of like a Cabinet War Room, but for the Navy. If you have a spare day take a daytrip to Chester, only 45 minutes away and a beautiful English town.

A nice short day or half-day trip is to take the train from central Liverpool to the town of Port Sunlight on the other side of the Mersey and visit the Lady Lever Art Gallery, which houses the art collection of the Lever family, who owned the "Sunlight" soap factory for whose workers the town was built. It's particularly good on Pre-Raphaelite art and Wedgwood pottery, but has an excellent collection in general. Bebington railway station is closer to the gallery than Port Sunlight station itself, but they're on the same line.

If you like cathedrals, Liverpool has two very imposing ones. The world museum is great too. I would second a train to Chester, a very picturesque and historical small city.

If you let us know if you have any particular areas of interest we can give you more specific suggestions.
Merseyrail trains run across the city and further afield, as far south as Chester and North to Southport also onto the Wirral peninsular so you will have a lot of options.

Agreeing with Philip above Port Sunlight is worth a visit if you have an interest in art and architecture. The story is that when planning to build a village for his workers Lord Leverhulme held a competition for designs for the cottages. Instead of going with one winner he basically decided to use all of them so there is a real range of late 19th century arts and crafts architectural styles. It's very pretty and well looked after. It's not very big, and is nice to wander around on a sunny day. My dad worked there for Unilever for years, so I have a real affection for the place.
https://portsunlightvillage.com/

I agree with Laura about the National Trust tour. It was the highlight of our very enjoyable time in Liverpool.

Yes, Forthlin Road and Mendips are unmissable if you like things Mop Top.