Galio's early game is pretty basic. You have excellent wave clear potential with your Q and your passive auto attack proc. You want to focus on poking the enemy and farming simultaneously. You want to try and aim one side of the Q at the minion wave while trying to land the combined gust at the enemy for max poke. If you can pick up a Dark Harvest stack or two before first back you are off to a great start. Make sure you are warding before 3 minutes as you are very prone to being ganked. I always like to keep my E off cooldown to get away from the inevitable early gank.
I like to try and back with enough gold for hextech revolver or 1200-1300 gold. If you back with less you can always make sure you have corrupting potion, and dark seal in preparation for the Mejai's build.

As soon as you have built your Hextech Protobelt you have hit a nice power spike. You can now use your active item to help you farm, lock down more Dark Harvest spikes, secure kills, and even escape death.
Use your ult to roam bot lane and try a few ganks. Preferably after you shove the wave to the enemies mid turret. If you secure a kill or two, look for the early dragon with the team, or get yourself back to mid lane to continue farming.
If you have Dark Seal and you have managed to score some stacks, think about picking up Mejai's, but I wouldn't do this without stacks as it is a gamble. Look to buy Luden's next as this burst damage with allow you to bully your mid lane opponent and put pressure elsewhere on the map by roaming.
The hardest part of playing Galio well is using your ult in a way that is always beneficial. Instead of ulting on top of someone who desperately needs your help, you may want to ult on the teammate farther into the fight as your CC will effect more enemies. That's my rule make use of your abilities in a way to most negatively effect the enemy team.
Galio is somewhat dependent on your teammates, that can be said for most champions, but when you have a beautiful 5 man ult lined up and your dumb *** jungler flashes away from the fight at the last second that can tilt you all the way to downloading Fortknive again.

When it comes down to it you have to be the one with the biggest set of balls on your team. You have the damage to one shot carries and the CC to protect your own. Make your decisions wisely, and judge what you want to do on the performance of your teammates. I have lost many a game with a diving 4 man taunt and almost killing the entire enemy team, BUT my team was too consumed trying to peel the one person on the enemy team with balls. If you have a someone on your team with bigger balls then you, that's fine. You wait for that large sacked bastard to dive the enemy and you ult on top of his head bringing the wrath of justice with you.

Galio is a great early to mid game carry so be aggressive and win your games by smashing the enemy so hard they turn on their jungler and he starts running it down mid. Be all over the dragons and rift heralds as season 10 will prove to be an objective based games. And the way things are panning out you don't want games lasting over 30 minutes as all the 1-12 Dr. Mundo mains will be able to run through your team and stall it out to 40-50 minutes where Galio just becomes a meat cushion for the enemies sack taps.

I am currently streaming on Twitch, come check me out and watch me rage and complain that other people are ruining my games, when it's definitely my fault.

Find me, Calabeast.

Thanks for checking out my guide, I hope you learned something, and even if you didn't hope you had a laugh.