Short answer: Park once in Dartmouth town and use the boats. The road out to the castle is narrow and the parking there is minimal, so take the ferry instead. Blackpool Sands, fifteen minutes west, has its own paid car park with a tricky final approach road.
Dartmouth is a river town and the boats are actual transport rather than a novelty. Ferries run downriver to the castle, across to Kingswear for the steam railway, and upriver towards Totnes. Park once in town, then move around on the water — that is the difference between a relaxed day here and an afternoon spent reversing down a lane.
Dartmouth Castle is a mile from the centre and the road out to it is narrow with very little parking at the end. The little ferry from town is the recommended approach on almost every review, and it gives you the best view of the castle as you arrive. Walk back along the water's edge afterwards if the weather is kind — it is a pleasant half hour.
The family beach is fifteen minutes west and has a decent car park you pay for as you leave, at around £7 for the day. The catch is the final stretch of road down to it, which is steep and narrow and catches people out. It fills on hot days, so aim to arrive before ten.
Worth knowing before you book. The Floating Bridge, by the higher ferry, has three parking spaces on the kerb — there is more parking nearby but do not plan on pulling up outside. Most of the good eating in Dartmouth is in the narrow streets behind the quay, so accept that you will walk from wherever you leave the car.
Strete Gate and Torcross, further round Start Bay, have their own parking and are far less busy — but check for road closures first, as storm damage has closed the through road at times.
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