Hello Cherryplanter, and welcome to the forum!
Coming from California I kind of feel the same way but Italians do things the Italian way.
The closest beach to both Pisa and Lucca is Viareggio which easily reachable by train (15 or 20 minutes). The main beach section of Viareggio is the usual collection of umbrellas and chairs but immediately either side of that are free beaches - Spiaggia Libera. The one closest to the train station on the north end just before Porto Viareggio is unfortunately in front of the port is not terribly attractive as a background. The free beach to the south fronts the start of a long nature preserve by the beach so the shoreline continues uninterrupted for some ways south. I think there's even a nudist beach if you head down far enough.
It looks like you could take the train to Viareggio and the 35 bus for 12 minutes to the southern end of the beach but it would be easier driving. The address is Viale Europa, 59, 55049 Viareggio LU, Italy for that beach access.
There are free beaches in Venice but nearly all beaches are out on the Lido so it will be the same deal - take the vaporetto out to Lido and then travel to the ends of the paid beaches to get to the free ones. I've never been to the beach in Venice but check out Murazzi Spiaggia libera as one possibility. All the best beaches are right in that area.
A quick tip when looking for free beaches in Italy is just use Google maps satellite view and look for where the organized umbrellas and chairs patterns suddenly break into chaos - that the free beach.
Hope that helps, have a great trip,
=Tod