You're best bet is the Tunnels -
The Tunnels Aberdeen, independent music venue & restaurant
They do acoustic nights as well - and they're the only place who will give folk a gig no problem really.
Other potential options are quite limited:
Drummonds -
MySpace.com - Cafe Drummonds - Aberdeen, UK - Other - www.myspace.com/cafedrummonds
A couple of places have closed/are closing lately, but other places to try (I don't know address off the top of my head) are the Moorings, Club Snafu, and possibly Moshulu (but you'd really need to be supporting someone fairly established to play there as it's massive!)
You'll get these guys myspace addresses generally from our top friends at:
MySpace.com - The Underkills - UK - Rock / Alternative / Indie - www.myspace.com/theunderkills