Port Sunlight (15 minutes by Merseyrail). William Lever's model village across the Mersey, built for his Sunlight Soap factory workers in the 1880s. The Lady Lever Art Gallery alone justifies the trip, and the village itself — Arts and Crafts cottages on planned curving streets — is a UNESCO-tentative-list site.
Chester (40 minutes by train). A handsome walled Roman and Tudor city to the south, with a complete circuit of medieval walls, a half-timbered shopping street (the unique two-tier Rows), and a Roman amphitheatre. Easy half-day from Liverpool.
Crosby Beach and Another Place (25 minutes by Merseyrail). Antony Gormley's 100 cast-iron figures gazing out to sea on a wide sandy beach. Combine with lunch on the seafront.
Manchester (50 minutes by train). Liverpool's old rival, with a different industrial heritage and a rich music history of its own — the Manchester Art Gallery, Whitworth Gallery, Northern Quarter and the football museums.
Speke Hall (40 minutes by bus). A Tudor half-timbered manor house six miles south, in National Trust care, with a fine Great Hall and gardens.