You don't say when you're going, but you might wish/need to be aware that from Thursday 16th to Sunday 19th of July, it's the Open Golf Championships at Royal Birkdale. This will make Liverpool ~ Southport trains - and Southport Town Centre - very busy. Similarly - but to a lesser extent - the 'Practice Days', Sunday 12th - Wednesday 15th July. This will have a knock-on effect on
all Merseyrail timetables (not just Liverpool-Southport), especially for the 'competition' days, 16-19 July.
One suggestion: Buy a Merseyrail 'SaveAway' ticket (£6.80) - which covers all public transport in Merseyside County. Do a one-way trip across the Mersey on the ferry (it's majorly advertised as a 'River Explorer Cruise' at £14, or £12.50 for concessions - but
a SaveAway ticket is valid providing you're only doing a one-way crossing, so Liverpool to Seacombe
or Seacombe to Liverpool -
link here).
From Seacombe, get a bus to New Brighton - every few minutes, or if you're feeling energetic, it's a flat walk along the promenade - about 3km. Lots of cafes, pubs etc in New Brighton. Then train or bus back to Liverpool (trains every 15mins, buses every 10) which will take you through the rail or Wallasey road tunnel repectively - or bus 410 (every 15mins) from New Brighton to Birkenhead Park (
link here). From here, there are very frequent buses to Liverpool through the Birkenhead tunnel, or 8 trains an hour from Birkenhead Park station.