Wascally is right about the main arterial routes out of Plymouth and whatever you do, I hope you relish hills.
Plymouth is a sprawling city but once you get out of it, there are some beautiful bits of countryside.
Going west, you could take the Cremyl ferry across to Mt Edgecumbe and sample the delights of Rame, Cawsands, Kingsands or make a longer loop down to Fowey returning on the Torpoint ferry.
Going east is a bit trickier as you have to navigate your way through Plymstock/ Elburton but once you have got out of the urban sprawl, the South Hams will open up for you. A loop out to Kingsbridge is recommended as minor roads are often an option. You could even combine this with a visit to Dartmoor.
Going north there is the Plym Valley cycle route which if you are on a mountain bike can be followed all the way to/from Princetown (in the middle of the moors). On a tourer/hybrid you could easily follow it all the way north/south on the stretch between Yelverton and Plymouth - the main 'rough stuff' being limited to Saltram and between Yelverton and Clearbrook (which is avoidable by picking up the road from Clearbrook onwards).
Bomber of this parish is ensconced there and may have other ideas.
Have a good one and say hello to the Dolphin on the Barbican from me.
H
