Well the ASPC site and Aberdeen property leasing will find you stuff when you know where you want to live. Other places to have a look at are Rosemount, The West End, & Ferryhill as well as places that are in the town centre. Nothing wrong the Bridge of Don, but I wouldn’t choose to live there.
Other places to avoid are Mastrick, Northfield, Woodside and to a lesser extent Kittybrewster & Hilton
There are plenty of other nice areas in and around the town (e.g. Mannofield, Cragiebutler) but they are unlikely to have flats for rent and any houses there are likely to be more expensive as would place on Deeside.
If you know where you are working then that will inform the better places to live. e.g. you don’t really want to be living on the south of the city if you are working in around Dyce and similarly a communte from Bridge of Don to Altens wouldn’t be a lot of fun either doruing the morning rush.