The scheme is for visitors to the council's ten libraries, two mobile libraries and four heritage sites and to encourage new visitors.
"It couldn't be simpler," said Cllr Adrian Blackshaw, Herefordshire Council's cabinet member for economic development and community services.
"All you have to do is collect a loyalty card when you visit any of Herefordshire Council's libraries or heritage sites.
"Each time you visit you'll get a stamp and after ten stamps, adults will get either a voucher for the Courtyard Centre for the Arts or one that can be used at any Odeon cinema, including the one in Hereford.
"Every adult will also get a free Bag for Life.
"Children will get a prize after three stamps, another after six and a bigger prize after ten.
"The final prize will be free child's admission to Shortwood Family Farm or the Small Breeds Farm Park and Owl Centre in Kington, or a Halo voucher that can be used for free swimming or a variety of other activities at any of the centres throughout the county.
"This is a unique chance to visit and enjoy our free museums and libraries and get a prize that opens up our other cultural venues. It's a fantastic way to get that cultural habit," he added.