Galio is an amazing champion and i would like to share my thoughts and opinions on how he can be played. Hes is one of my favorite champions and definitely my main so i do have some qualifications.
I will be improving this over time adding more detail to the dialog, but i cant really fully figure out how the adding pictures and links work(i can only add pictures for now) so i will have learn that from someone at some point. Until then i will just have to work with what i have, getting the words to about how long i want it to be like My Kennen guide.

This is the must have summoner spell after all. Since it can help you get out of sticky situations, dodge abilities and gap close on enemies. And it will help you get some great engages with your W but not your E since you can redirect it. If you can use it properly you can easily get a kill for your team.
Extremely useful because it can let you spend everything and back then instantly get back into lane or it can help you assist your teammates in many different ways.
Since most people dont really understand how much damage Galio can do even after the nerfs your full combo will take players but surprise but it is not enough to 100-0 someone. Thats where Ignite comes in. It will help you secure the kill for sure but you do have to nib and lick at your enemy first, get them in a spot where they feel safe enough to not back(about 60% will do on that squishy mage or that yasou player).

AfterShock-This is one of the best choices for Galio as a Keystone. It offers a lot of resistances and a bit of damage so its ideal for a champion that has a lot of cc and who wants to be the one engage or follow up on one. But the resistances are momentarily so dont be too confident.
BonePlating-It takes a some damage away from 3 abilities so its generally a great rune. In lane it makes it so that bursty champions like Zed or Ahri cant even touch you anymore as well as slicing some damage off when their jungler ganks.
Chrysalis-This rune is designed to help early game champions but it gives you AP so thats the main reason to go this. It is a bit nice to have a bit more HP early on but you wont really notice the effects too much.
Conditioning-The must rune for every tank because boosts your resistances and gives you that juicy bit of MR that i like so very much. Its useless unntil the 10 min mark but thats not really problem for Galio since he still has a decent early game.
OverGrowth-Gives you about 2-4 health crystals worth of health. Any extra health is great to help you survive a bit longer. But i mainly go this rune since Galio has no heals of his own to take Revitalise and i dont really like Unflinchable.
SuddenImpact-Gives you some nice bit of MP when you proc it with your E so it will over all improve your damage output and will make your all ins that much scarier.
UltimateHunter-The best of the hunters. This is the main reason you go Domination secondary since the nerfs on Galios ultimate cooldown and the removal of UltimateHat. Unlike ultimate hat you dont really need to have your ult to stack it, if you can get kills pre level 6 it will still stack, but its rare to get all 5 stacks in the early game so it wont really fully stack until you can get in some team fights.

CollasalSmash(Passive)-Makes your autoattack have an Aoe affect it adds to Galios wave clear and since it scales off AP it will add to your burst. If you can land all of your abilities in a fight it can proc twice because the cooldown gets lowered by every ability hit.
Wind Of War(Q)-This is Galios main form of wave clear and damage. It has a pretty short cooldown and a not too bad mana cost you can definitely spam it in lane to harass you opponents and once you get a bit of AP it will clear minion waves in a beat. It has initial damage and over time damage so when you combo your enemy CC them with your other 2 abilities so they stay in the area of your Q and take the full damage. A Basic combo is to W or E into landing your Q and using the last ability to make them take the over time damage then finish it off with your empowered auto attack.
Shield Of Durand(W)-An Aoe taunt around Galio that you have to charge and it has a passive that gives you an magic damage shield. This is why you can counter mages so well and it also gives you damage reduction during the channel, taunts the enemy based on how long you charged it as well as doing quite a bit of damage. It also cant be interrupted by CC so it makes it really hard to gank Galio since he can just taunt you even if you stun him and just dash away or if he is under turret or a teammate is near by its oh so easy to just turn on the enemy. As mentioned you can also combine this with your flash to get a really good engage(charge it fully before you flash for maximum effectiveness).
Justice Punch(E)-Simple dash ability that does a fair bit of damage and will knock up the first enemy hit. This will be your gap closing tool or it can help you combo with your taunt but it will stop on the FIRST enemy is infront of you so someone can block it for their teammate and it a bit hard to escape with it if they know how your E works so be sure to W before so you can get a good position. Also you cant go over walls with it.
(Fun Fact: Before people used to max this ability second but after they made it so your W did damage it got outranked.)
Hero's Entrance(R)-One of the best team fighting ultimates out there since it gives damage reduction to the area, delivers Galio to a teammate(Cant ult yourself!!) and Knocks up the enemies in an area when He Brings Down the Thunder. This is the ability that makes him the HERO. But its a bit less usefulll since they nerfed the cooldown on it. This main reason why you play him mid so he can affect the map better(Dive side lanes, Help your jungler invade since you can respond first, Turn fights to your advantage or ensure a fight). If you cant reach someone E first.

Early game:
Poke out you opponent and farm and push as much as you can(Try not to poke your enemy out of lane if you want all in and try to get a kill). If the jungler ganks just W and E away you shouldnt really need to blow a summoner. Oncce you get level 6 find an oppertunity and do your full combo(E>Q>W>AA) even if this wont kill you will push them out of lane. If your jungler ganks for you charge your W and flash taunt into Q and E to get a guaranteed kill. Try not too back too much and push push push until you can get your Ultimate.
Mid game:
Push and roam. When you get your ultimate look at the map carefully and make plays. Especially your jungler, watch them carefully and make sure you can follow up on their ganks or when they invade since you can respond first. But make sure your enemy mid isnt just free farming or ganking the opposite lane without warning when you roam.
Late game:
Team Fight and Push side lanes. Since you have so much wave clear and you can escape quite easily but still enter fights from far away. With how much CC you offer alone you can follow up and secure a fight or turn a fight completely to your favor. When ever you do a massive engage and win a fight take that lead and turn it into something like a baron or an inhib.

Galio is an amazing champion who can truly save his teammates and protect everyone.
After his changes he became more of of a mage but he still can never be a true mage so he can very flexible. If you are ahead of your laner go AP to snowball your lead and if you behind go Tank.
He is quite easy to pick up but to maximize your kit you will need to roam and affect the map as much as you can. I would recommend you watch your jungler as much as you can.
This is my first guide so its not as good than the other guides, i will be improving this guide over time as i learn the system more.
I hope you found this guide useful and good luck on your games.