Short answer: The Box is free, including its 4D history film, and closed Mondays. The Hoe, the Barbican and the Mayflower Steps cost nothing. Devil's Point gives you three hours of free parking with accessible paths and benches, which is the best free start to a day in the city.
Free entry to Plymouth's main museum and gallery, with a 4D film on the city's history that is the part children remember. It is closed on Mondays. For a free museum it is unusually substantial — this is not a room of cabinets, and it will take a couple of hours if you let it.
The great open green above the Sound, free and always open, with Smeaton's Tower standing on it. It is the view that explains the city: the breakwater, the naval dockyard, the water going out to Cornwall. You can spend an hour here without spending anything.
Cobbled streets, galleries, the Mayflower Steps and the working harbour. Wandering it is free; the boat trips that leave from it are not. The cobbles are rough under a buggy, so keep to the waterfront edge.
Three hours free parking, accessible paths and benches, sea swimming access and a clear view over the Sound. Free parking in a city is worth as much as a free attraction, and this is the one spot in Plymouth that gives you both.
Walking round the converted naval buildings costs nothing, and the scale of them is the point. There are restaurants, bars and a cinema if you want to spend, and a ferry across to Cornwall in season.
The Lion's Den, below the Hoe, is popular in warm weather and free — though it is heavily graffitied and the access down is awkward, so plenty of people are content to look at it from the Hoe above.
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