Short answer: Plymouth is a city on the water rather than a beach town. Devil's Point has good sea swimming access with accessible paths and three hours free parking. The Lion's Den below the Hoe is the other in-city option — popular when warm, and heavily graffitied.
Plymouth is a great waterfront city and not a beach resort. There is no long stretch of sand in the city itself. What there is instead is sheltered water, swimming spots cut into the rock, and a Sound that is protected by a mile-long breakwater — which makes the water here calmer than the open coast either side.
The best in-city option. Good sea swimming access, accessible paths and benches, and three hours free parking. It looks straight out over the Sound. The water moves here, so it suits confident swimmers rather than paddling children.
Cut into the rocks below the Hoe, free, and busy whenever it is warm. It is heavily graffitied and the access down is awkward. Plenty of people look at it from the Hoe and decide that is enough, which is a fair judgement.
You go east or across the water. The south Devon beaches at Bigbury and the South Hams are a drive east, and the Cornish side is reachable by the seasonal ferry from Royal William Yard. Within the city, treat the water as something to swim in or look at rather than to spend a day on with buckets and spades.
Mount Batten and the Sound's eastern side give you water with less graffiti and fewer people than the Hoe, and the boat trips from the Barbican are the best way to actually be on the water if swimming is not the plan.
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