I'd recommend a visit to Tentsmuir Forest near Leuchars. It's a National Nature Reserve on the Tay Estuary. See
here and
here for more details. I visited last summer and although it was packed with people very few ventured into the forest (they were all BBQing on the beach and car park). Corsican and Scots pine make for an interesting mix with some interesting wildlife.
As previously mentioned Loch Leven is also worth a visit (
RSPB Vane Farm reserve or
NNR here) or how about a trip to the
Isle of May? Boats go from Anstruther.
Loch Garten and Abernethy Forest Nature Reserve (Ospreys and Capercaillies respectively) are also just about "do able" from Dundee but it would be a very long day. You could try to go up Ben Chonzie near Crieff? Fairly easy access from Perth/Dundee
see here for details.
Hope that gives you some ideas.