You don't need exalted reputation with any faction to get flying training. Technically the ability to use a normal flying mount (60% on land, 150% in air) is just having Expert Riding skill (225). You'll 600 to 650 gold for the training and the mount. If you want a swift (100% on land, 280% in air) flying mount, you will need 5000 gold to get to Artisan Riding skill of 300. To fly in Northrend you will also need an extra 1000 gold to get Cold Weather Flying from trainers in Northtrend.
Riding trainers that teach flying: