I have chosen this small part of Plymouth because it is Old Plymouth. It is the site from which the Pilgrim Fathers set out in 1620, and it is the part of Plymouth that retains some charm.
There are a number of things to do here. We found it easy to park on the east side of the harbour near the National Marine Aquarium. The aquarium is large, and since we were there, it has added even larger tanks, so it is well worth the visit...a great place for both kids and adults.
From the aquarium, one can cross the mouth of Sutton Harbour on a footbridge. If you're lucky, a boat will cross when you're there and the bridge will "swing" into action. On the opposite side you will find the barbican.
Having crossed over, you will immediately be drawn to the Barbican Glassworks... it is very attractive and contains a wonderful showroom for blown glass and other artworks. There is a large variety of merchandise at all price points.
An interesting industrial tour may be the Coates Black Friars Gin Distillery. It features guided tours at specific times.
There are a number of other sites in the Barbican including St. Andrew's Church and the City Museum and Art Gallery, so it is possible to spend a day here. I can only suggest that if your time is limited in Plymouth that you start in this area (which is really quite small) and work your way over to Plymouth Hoe or into the commercial zone (great stores) if you have time... both are just a short walk away.