World Museum, Liverpool
Five floors, free, and it covers three things small children reliably love: giant dinosaurs, an aquarium and a space gallery. The best wet-weather day within half an hour.
Merseyside Maritime Museum
Liverpool, Merseyside
Titanic and Lusitania told in the city that crewed them, ship models by the hundred, and in the basement the Seized! customs gallery where children hunt smugglers with unnerving enthusiasm.
Free, and the Albert Dock outside is its own event - combine with the Museum of Liverpool for a full free day on the waterfront.
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Also in Liverpool
Speke Hall
A Tudor hall with woodland walks, a maze and a natural play area in the grounds. Free if you have National Trust membership, and the indoor house is a good rain fallback.
Museum of Liverpool
The city's story with an overhead railway carriage to climb aboard and Little Liverpool, a dedicated hands-on gallery for under-6s. Free, on the waterfront, and the window views are half the fun.
Mersey Ferries River Explorer
The famous ferry across the Mersey run as a 50-minute sightseeing loop with commentary, and the honest truth is kids love the boat more than any museum. Hop off at Seacombe for Eureka! next door.
Calderstones Park
South Liverpool's favourite park: lake, playground, ancient trees and the Storybarn, The Reader's hands-on storytelling space in the old coach house. The neolithic Calder Stones are older than Stonehenge, which is a good fact to deploy.
Croxteth Hall & Home Farm
A working Edwardian estate farm with rare breeds, a walled garden and 500 acres of free country park around it. The farm entry is cheap and the pigs are enormous.