Hi, there a few places. From the main barbican area, there are dedicated boats, that do dockyard and warships trips, which take you up and down the Dockyard. Note these are seasonal and do not operate in winter.
If your driving there's the main ferry between Plymouth and Torpoint, which is the opposite side of the Tamar River. The ferry is situated between North & South Yards of the dockyard. Across the other side of the river at Torpoint there are some good viewing points, but a 250mm + lens is recommended.
http://shipping.fotopic.net is my shipping fotopic, as i live in Plymouth to give you general idea of pictures you can get.
http://shipping.fotopic.net/p62926101.htmlhttp://shipping.fotopic.net/p63134868.htmlThats a couple of examples of picture taken on the ferry crossing.
Otherwise you can get fantastic pictures from the main Hoe area of Plymouth overlooking Plymouth Sound.
http://www.thisisplymouth.co.uk/shipping Is usually pretty much spot on with the movements of naval ships. For ships coming in and out in the sound, the best spot for photo's and viewing is a place called Devil's Point.
http://www.streetmap.co.uk/place/Devils_Point_in_City_of_Plymouth_356611_64611.htm A bus service does run down to there, but i would recommend use of a car as its a bit of a walk from the bus-stop.
Heading out of the dockyard
http://shipping.fotopic.net/p63238868.html Heading into the dockyard
http://shipping.fotopic.net/p63199355.html Hope this helps
All the best
Allan