Which building to upgrade first?

All buildings are important to upgrade and in the beginning of the game at least you should be spending all of your speed ups and supplies on getting as many upgrades done as you can... Why? Because it unlocks more stuff, increases your player level, and increases your power in the game.

It's not really important which building you focus on. But pay attention to the milestone improvements where something extra is unlocked, for example, more units in a barracks, certain research items that you need to increase your barracks level, an extra production queue etc...

Targeting these milestones, and upgrading certain buildings to reach these is a good plan to have.

Take for example the production center here you get an extra production workshop at levels 7, 17 and 22. So efforts should be made to reach one of those milestones before focusing on another building that has their own milestones.

To find out how upgrades affect a building, tap the upgrade button, then tap the little 'i' that will give you a table of information of all the building stats for each level.