It looks like you are in Liverpool for a few days. One station I would recommend is James Street (central Liverpool) on a weekday. It has two entrances, one on James St, that uses large passenger lifts to take you underground (very like London Underground), the other, on Water Street is only open weekdays between 7-10am and 3-6pm, it goes down some stairs, then a long ramp with handrails and restored original signage. Unusual.
Platforms 2 and 3 are on the original Mersey Railway station whereas Platform 1 is part of the 'Loop line'. Platform 3 is modernised in Merseyrail underground style, Platform 2 is rarely used and has not been modernised, but brought back to a pastiche of an olde station. It is now dimly lit (usually in purple) and looks a fitting place for a ghost train! Very atmospheric! You can see it from Platform 3.
Southport line
Once out of the underground tunnels the line runs mainly on viaduct/embankment giving views over industrial/commercial areas and the docks and River Mersey until reaching Seaforth.
Further on there's lots of overbridges good for photography. Mersey Road, close to Blundellsands and Crosby is good for shots in either direction. An interesting one is further down towards Waterloo at Manley Road/Somerville Road, a footbridge with good views of the track...the bridge is built of steel plate and has bullet holes through the plates from when a Messerschmidt shot up a freight train in a low level attack in WWII!
If you get off the train at Waterloo, Blundellsands or Hall Road then you can take in Antony Gormley's 'Another Place' on the beach.
en.wikipedia.org
The road overbridges at Formby and Hillside have near identical station facades onto the bridge, with original L&Y decoration. More photo opportunities of the trains from here.
Between Freshfield (alight here for red squirrels) to Ainsdale the line has heathland on one side, followed by an RAF base
(edit: and a golf course, open grass fields then pine woodland on the other side).
Edit: Between Ainsdale and Hillside there are golf courses on both sides of the track!). In fact, there are golf course all over this area, with several visible from Merseyrail.
Ormskirk line
Walton and Orrell Park have good views along the track (and are very close together) and can make a good shot looking towards the other station.
Aintree has a road overbridge and a passenger footbridge for photos. Aughton Park is in a wooded cutting, often full of birdsong, very different from other Merseyrail stations.