Southport line is my fave. Inner city, coast and countryside with some fast running at the top end too. You may get a light aircraft landing at RAF Woodvale right by the side if the track if lucky. Also if there's an exercise on you sometimes get helicopters around Hightown (3 chinooks seen in July!) There's still bits of jointed track too, most notably Formby-Hightown-Hall Road southbound.
Red Squirrels about a 10 minute walk from Freshfield, the metal men on the beach at Crosby, again about 10 minutes walk away from the station.after Sandhills southbound I always enjoy playing "spot st John's tower/Anglican Cathedral/The Wigwam" a game my Grandma started off when I was a child.
If ultra lucky you may see a freight train running to/from the docks on the short space of parallel line between Oriel Road and Bank Hall.
Sandhills a good place to see everything on the Northern, Hamilton Square for the Wirral. Make sure to go past Kirkdale depot (Ormskirk/Kirkby) or Birkenhead North (West Kirby/New Brighton) to see what's hanging about. WK, NB and Rock Ferry may have stuff in the sidings. Also Southport sidings has quite a few units in attendance duringbthe day but have binoculars with you if you don't want to walk round to the bridge. They start strengthening southbound trains from Southport from about 3-3.30 so if you want to see some shunts and trains joining up with each other be there around then. Similarly the other way round after the morning peak about 9.30. It's the only place you'll see that happening on the network unless under special circumstances
I like the creepy ambience of Green Lane (complete with "Do not alight here" signs before the platform begins that used to face you in the very back of a 6 car set BY AN OPEN DOOR!) Hightown Station nice and peaceful. Check out the ultra thin platform section at Cressington too. You can see parts of the historic lever brothers workers village between Bebington and Port Sunlight. Several golf courses near the line on the way to both West Kirby and Southport too. If you see a 507/8 on the Kirkby line DROP EVERYTHING AND CATCH IT they will be physically barred from getting past Fazakerley after Headbolt Lane opens.
Try the new trains with an open mind. When they actually work they are perfectly acceptable trains - I certainly find them a much nicer environment than Northerns woeful 195/331 offerings but that's just a personal opinion. If they last even half as long as the old guard they'll build up a following in time. The stars of the show are the 507/8s obviously though. 508s more endangered than the 7s (and they still go ticka-ticka-ticka like the older emus!, the 7s actually have newer compressors so don't) So grab as many as you can. I've spent my entire life travelling on them, they were introduced the year before I was born so they run through my veins and the thoughts of them not being there any more still induces panic if I think about it too much! Please be aware though, the seating/window alignment is woeful!! (Didn't matter as much with the old low back seats but they're high back ones now)
For the extra nerdy element if you get off the Wirral line at Liverpool Central check out the OUT OF DATE network map on the up escalator. It's pre Liverpool South Parkway/Maghull North but post Brunswick.
That's just bits I can think of. There will be lots more, it's a really interesting network to ride around on with friendly staff too. Hope you'll enjoy your time in the area. (And whilst above ground check out the waterfront near Albert Dock, on a sunny day its a really glorious place to be)
Best wishes