You will have a lot of different buildings to upgrade. Initially, don't even worry in the slightest about what they each do, just tap through as the game tells you to in the overlong tutorial.
Once you've gotten through the tutorial, check through all of your buildings one by one, and use any free upgrades you have on each. Believe me, it'll help in the long run.
Don't bother using the speed-up items quite yet, there will be time for that. Feel free to look at other games or other aspects of this game while everything ticks over and upgrades… Or, y'know, watch ads for bonuses.
Several of your buildings won't be there just for upgrading your troops or vehicles or whatever, some of them are there for cold-hard research, and you'll need to oversee that.
As they research and advance, you'll be choosing what skills from research trees you want them to focus on.
You'll initially be able to choose between Arms Type, Ruling, and Import at the Research Lab, and if you want military might, go for Arms Type and upgrade that tree.
Shooting nano zombies
This is where things get kind of dumb. Sometimes, seemingly randomly, zombies will start pouring out of a tunnel near your military base. Again, I'm not sure why, it just happens.
It's a good excuse for an action scene though, and you'll take to a chopper and start licking off machine gun shots from the sky.
The "trick" here is to hit the explosive barrels to make light work of them. Hey, at least you finally got to go hands-on with an actual weapon.
One more shooting section, this time it's terrorists. Terrorists will seemingly spawn and live in the streets of your military base without any interruption, until your interrupt them with a sniper bullet.
You'll be doing this from the top of the primary tower, and… well, yeah, you'll snipe terrorists. Aim down your sight and tap away to lick off bullets, don't worry about bullet drop.
Is this totally pointless? Do the terrorists just seem to be minding their own business? Is it weird that the dozens of soldiers you hired wouldn't do this themselves? Yes, it's all very odd, but shoot them anyway, I guess.