My opinions:
Baku is a quirky and certainly one-of-a-kind city. There's a weird mix of things here, from some medieval relics to some lovely Csarist-era architecture to all kinds of crazy oil-related sights, like burning hill-sides and mud volcanos. It's an odd place to spend a few days, but can certainly be interesting.
Tbilisi is the loveliest city of them all, by a mile. The old city is well-preserved, the Soviet ugliness is kept far from the city center, and the city is ringed with pleasant hills. There are also some interesting day-trips, and plenty of delicious Georgian food. Spend your most time here.
I didn't like Yerevan all that much. I found the city ugly, and oppressive Soviet architecture certainly dominates. The most interesting thing here, in my opinion, is a trip to the nearby town of Echmadzian, which has been the center of the Armenian church for centuries and centuries. I would spend the least time here if I was you; a day to take in Echmadzian and anything else you have time for should do, given your rather limited time scale.