Hop on a train (or the ferry if its running) and get off the other side and ride along the prom around the Wirral.
(The Wirral is good for cycling in general, lots of little lanes)
Go to Widnes on the Trans pennine trail (N62), then along the canal (still 62....) nice place to turn around is the pub half way.
The bit in Widnes is a bit confusing atm, but there are signs. Once you follow the path through Pickerings pasture, go under the bridge and along the river to the Catalyst. You used to be able to go on the canal path there, and hopefully with the new bridge complete they will restore that, but for now you have to go onto the main road and off where the ice rink is. There's a path to get to the canal there. Its got two massive concrete blocks in front of it and looks like its going nowhere, but if you follow it there's a gated railway crossing and a bridge over the canal.
Follow the 62 out North to get to Southport without going on the road.
To be fair I do most of my cycling outside of town
Oh, sustrans will have maps, and merseytravel might have some too. I picked a couple up for the different boroughs, but that was a few years ago.