In reply to Ste Brom:
I reckon there is no city centre in the UK with a three star guidebook route closer (thinking of Talisker at the Whisky Cliff, approx 1.5 miles from the Bon Accord Centre).
Aberdeen is without a doubt the best city in Scotland, and one of the best in the UK for evening cragging, with over a thousand routes on dozens of crags all within an hour's drive.
The stretch between Nigg and Cove is easily accessible using one of the city bus routes. Newtonhill is also on a bus route, and from there, it's only a couple of minutes walk to the nearest crag.
Logie Head, Clach na Beinn and the Pass of Ballater are the pick of the crags away from the coast once you get yourself sorted out with transport (though you could easily do the Pass by taking the bus to Ballater itself and walking round to the crag).
Get yourself a copy, or at least a look at Northeast Outcrops - it'll show you what's what.
And Norrie - I'd ease off on Mike C. At least he's an Aberdeen local and is being helpful. There are very few Aberdeen locals posting much on Scottishclimbs at the moment, so I suspect the OP has less chance of as much useful advice posting on there. In my opinion, of course.