Short answer: Start on the Hoe because the view explains the city. Then pick your indoor stop by the day — the Box is closed Mondays, the aquarium closed Wednesdays. Barbican for lunch, then walk west to Royal William Yard and finish at Devil's Point, where parking is free.
The Box is closed Mondays. The National Marine Aquarium is closed Wednesdays. Those two closures catch out more visitors to Plymouth than anything else, and they are easy to plan around if you know in advance.
Start here. The Sound, the breakwater, Drake's Island and the dockyard all lay themselves out in front of you and the rest of the day makes more sense for it. Smeaton's Tower is on the green — you can go up for a charge, by spiral staircase.
The Box if it is not a Monday: free, with a 4D history film, and substantial enough to take two hours. The aquarium if it is not a Wednesday: the UK's largest, and the ticket includes free re-entry for a year, which is worth knowing even if you are only here once.
The Village Restaurant does a good-value set lunch and knows its seafood — book, and expect noise. Fig & Thyme is the brunch option if you are earlier, though you will wait if you arrive late. The Barbican Kitchen, inside the gin distillery, is closed Monday and Sunday.
Walk west to Royal William Yard and look round the converted naval buildings, then on to Devil's Point for the view back across the Sound. This is the stretch most day visitors never reach, and it is the part locals use.
Not the Barbican, where it is expensive. Devil's Point gives three hours free and Royal William Yard has good parking. Park west and walk east into town, which also means the end of the day is a short stroll rather than a trek back.
Boat trips from the Barbican get you out onto the Sound. The seasonal ferry from Royal William Yard crosses to Cornwall. Dartmoor Zoo is a drive out but the annual pass costs about the same as a day ticket, so it rewards a second visit.
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