Despite the relatively small size, there's a lot to do in Azerbaijan. I recommend balancing your time between Baku and the countryside. But when I say countryside, I'm referring more to natural and historical sites scattered throughout the country. Azerbaijan has a huge, long, and rich history being one of the earliest settled places in human society and being at the crossroads between Persian, Turkish, and Russian cultures. Try to do and see things here that you cannot do elsewhere, such as the Zoroastrian fire temple Ashtegah, the Caspian Sea coast, and the country's mud volcanoes (since nearly half of the world's supply exist only within Azerbaijan). Open yourself to new experience and try to make friends. People can be incredibly welcoming here and that makes the trip infinitely better.

Recommendations

Visit Gobustan, preferably at a quieter time
Walk throught the medieval castle in the center
Drink traditional Azeri tea
Go to the Caspian Sea
Make friends (never have I been treated so well before)
Find a street musician
Make the trip to the fire temple
Prepare yourself for the wind
Absolutely bring your camera
Do not be in a car during rush hour
Try the shisha/hooka
Walk the many stairs to the top of the city