I'm a local, feel free to PM me for more info. For running near the airport, you can easily get onto an old railway line which has been converted to a long distance walking/cycling trail, see for example
https://www.walkhighlands.co.uk/...e-udny-station.shtml . If you end up in the city, or on the Southern end, there is the Deeside way, which - while flat - is a lovely run. It starts at Duthie Park, and you can follow it all the way into the Highlands.
For cycling, the North East is a great area, rolling hills and not much traffic off the main roads. There are several clubs you can go along with, including Deeside Thistle and Fleet Feet Triathletes (the latter also do turbo training sessions once a week I believe).
Others have mentioned the University of Aberdeen sports village pool, which is indeed excellent; it's usually 50m on Mondays and Wednesdays, and 25m on other days. There's also a lovely outdoor swimming venue -
Knockburn Loch, where you can also rent a wetsuit.
Let me know if you need more info...