Barcelona

For transport get a TMB 10 journey T-Casual ticket.
https://www.tmb.cat/en/barcelona-fares-metro-bus/single-and-integrated/t-casual

For museums the forum usually agrees that the ArTicketBCN is your best value option.
https://articketbcn.org/

There does not appear to be a useful, money-saving option for travelers primarily interested in the popular Gaudi sites.

indeed Ann, there are none for 'the Gaudí sites' (sic), not really useful anyway... the reason being that these are operated by different private enterprises (or foundations) and their attractions, as it is, are always fully booked, therefore there is no incentive to promote any scheme for reduced prices.

Thanks everybody

At the Museu Picasso, where visits are scheduled by time slots, there is a fast-track entrance for ArticketBCN Art Passport holders in Carrer de Montcada, and you do not need to make a prior booking. Show your ArticketBCN Art Passport to a member of staff before queuing so they can direct you to the ArticketBCN fast-track entrance.

Thanks David