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Sefton Park, including the palm house. Nearby refreshments include Gelato on Lark Lane for ice cream, Handyman's Supermarket and the Craft Taproom on Smithdown Road for beers.
Baltic Triangle for bars, crazy golf and the Baltic Bakehouse.
Albert Dock for Museum of Liverpool, Tate, riverfront. Don't know when you are thinking of visiting, but the Clipper yacht race is expected to finish there around July 28th.
William Brown Street for world museum, Walker art gallery, and the Central Library (pretty cool architecture inside, to my untutored eye at least). Ship and Mitre pub is nearby, a perennial Camra favourite.
Anthony Gormley's Another Place at Crosby beach, red squirrel spotting a bit further up the coast at Formby. Both easily accessible by train.
Any recommendations for stuff to do/places to eat etc in Liverpool? Beatles optional.