Mark,
Check out
http://www.derbyshireos.org.uk/
I grew up in Derby, travelling without your own vehicle isn't easy, however, if you are near the train station then you can get to Willington Gravel Pits easily. From the train station it is just a 300m walk along the main road, go under the rail-bridge towards Repton, walk down the dead-end lane on the right just before the river Trent, can't miss it. The pits on the left are private fishing pits, the ones on the right have viewing platforms and is being re-created by Derbyshire Wildlife Trust.
Some of my highlights from the pits include Pallid Swift, Bonaparte's Gull, Wryneck, Kentish Plover, Temminck's Stint, White Stork, regular Garganey, Pec Sand, Marsh Harriers, Med Gull, a flock of 158 Black-tailed Godwits then the odd Little Stint, Curlew Sand, wintering Jack Snipe and Water Pipit etc. Also first Derbys record of Small Red-eyed Damselfly, good site for dragons.
Last few years haven't been great but it could get better if DWT do the right things. As Redhank, Common Sand, LRP and Yellow Wagtails used to breed.
There are also regular Trent Buses that pass through Willington on the way to Burton. Carsington Reservoir (always stacks of waterbirds, few good birds, esp gulls) is reachable by bus, so would be good for a weekend visit (or weekday if your a student ;-) ).
Other sites that require your own transport include Foremark Reservoir (not great, but good for gulls in winter), Ogston Reservoir (winter gull roost as good as anywhere in UK), Shardlow Gravel Pits (Ortolan Bunting, Tawny Pipit, Arctic Skua, loads of waders, careful of access, Sunday pm only).
If I were you I would stick to the nightlife - attempted Derbyshire birding for 17 years with little success!
James