Name´s Faehana, I play on EUNE an EUW support. This is my first guide so I hope it´ll help you :)
I´m only lv27 on both accouts and I´m still learning. I started playing Galio after his rework because I liked his desing and voice lines and fell in love with his playstyle. I tried him once as a support and had lot of fun (we even won!) and while looking for good build for him I found out there´s not any in depths guide. So here we go.
(Please note that this build is still WIP)

Why should I play Galio support?

Galio as support is build to protect his teammates meaning he´s the one to go in fight and takes as much damage as he can. He´s like a living shield to hide behind. His spells offer nice CC and he deals enough poke damage to make enemy support cry. He is mana hungry early game but when played well not only you will win but you also will have fun. And that´s the goal of videogames, no?

PROS

+ Tankiness

+ Good CC

+ Good follow up

+ Nice poke damage

CONS

- Few spells activate after short delay

- Beside his E and Flash no mobility

- Easy to be poked by ranged support

Galio´s passive - Colossal Smash

This is his main minion wave clear mechanic when low on mana or just in general. I save it to wizard minions so my ADC can farm faster. It´s good to know it also works on towers so save it for that extra damage.

Q - Winds Of War

This is his second way to clear minions from safe distance while doing some poke damage to enemies. You can use it for zone control since tornado spins for 1.5s and enemies either have to take damage to farm or step away and loose some golds.
I max it last because as a support we should´t be doing damage but CC the enemies.
Definitely use it in team fights.

W - Shield of Durand

Going W first for his passive which is sweet early game when going against mage support or ADC.
When casting you should remember that W slows you down during first cast (also rember you have to hold your W!). When lettingW go or when charged to max you taunt enemies struck in the "circle" around you.
Funny thing I found out it triggers Aftershock! So use this tip to maximum during team fights.

E - Justice Punch

This ability is good gap closer but bad if you want to escape. Why? Because before dashing forward he takes a small step back. This can turn horribly wrong if enemies catch you.
On the other hand it´s nice knockup with decent damage and easy to land. Remember it only knocks the first person you hit.
I go E second to surprise enemies with suddent impact. JUSTICE!

R - Hero´s Entrance

Man do I love and hate this ultimate. It´s amazing gap closer with huge radius and knockup with sweet damage. The ally you chose to land on even gets damage reduction!
But it´s hard to land it.
As his E he has to charge before going in action and sometimes your teammate or enemies can simply walk away from the mark. Did I mention the huge shyny mark that also appears?
Besides that his R is really nice using for unexpected ganks or turning the fight over. It´s also really satisfying when you land it.
Play Video

Aftershock
This rune works well with Galio cause it can be triggered by 3 spells (W, E, R). The shockwave triggers after some time so you have to stay near enemies for them to be hit. This rune can be changed to Grasp of Undying.

Demolish
This rune isn´t use much but with Galio passive it deals nice amount of damage. It´s nice bonus early game and the damage you deal scale with how much HP you have (late game you deal a looot of damage). This triggers when under tower - you have to wait until all crystal under it expires then you will deal bonus damage. It has sweet 45s CD so you have it up whenever you get under tower after you win trades.
Can be changed to Bone Plating.

Second Wind
As a living shield you will take a lot of damage so this rune will give you some sustain

Unflinching
Tenacity is always good especially when going against CC based champions

Cosmis Insight
This rune speaks for itself. +5 CDR is always a good start

Magical Footwear
This rune gives free boots when hitting 10min or earlier (depeds how much kill you have). It also gives +10MS and small discount on tier-2 boots. Sounds good right?
If you want you can change this rune for Future's Market for money or Biscuit Delivery for sustain.

TANK

-Eye of the Aspect is the core to all builds here. When you collect 500coins with it upgrades and you get 3 wards. As a support you should put down ward whenever you think you might get ganked. The heal in passive isn´t much but it´s there. The coin share here is sweet.
-Rod of Ages in the second item you should build. It goes really well with Galio´s spells and becomes stronger late game. But if the game looks too quick to end you should go for something another because it has high price.
-Redemption may seems like odd thing to build but his W and Relic shield goes well with it´s passive. It doesn´t give a lot oh health but active is really strong area heal. You should remember it triggers after some delay
-Knight´s Vow make you more like living shield. It redirects some of the damage your Partner gets to you while healing when they deal damage
-Randuin´s Omen is a big middle finger to all AAbased champions who go Critical Strike. Altho it´s not much, it´s something. His second passi ecan be annoying to AA champions with Attack Speed and the slow on active can be game changing (it also has low CD)

ABILITY POWER

-Morellonomicom may seems like odd thing to build but it´s passive negates some of enemy healing and it gives nice AP and HP
-Luden´s Echo is another odd thing to go but 20%CDR and it´s passive is strong
-Rylai´s Crystal Scepter slow´s too good to be left out.

WHY ARE YOU STILL HITTING YOURSELF

-Abbysal mask gives you sweet bonus if you have team build around AP while it´s passive will help you sustain
-Frozen Mallet slowing enemies with your passive Colossal Smash is fun
-Thornmail is the definition of "Why are you hitting yourself?"

OPTIONAL

Galio´s build should be situational and these items can be used just as any used above

-Reason I max Shield of Durand first is for it´s passive against mages and it´s taunt
-If you wanna be more aggresive you can max Justice Punch first for it´s damage
-I max Winds of War last. If you´re playing him as midlaner or toplaner then you might argue but besides dealing damage it doesn´t give any supportive help
-Drop a point to your Hero's Entrance whenever you can

Flash is a must go to all champions. It´s good for gap closer or escaping fight
Ignite is starting to be favorite pick on supports especially melee. It´s best to use on low HP enemies either to negate healing or killing them with burn(Soraka or Sona)
Exhaust should be picked when enemy has strong AA damage and when they ADC has no mobility (like Ashe or Varus)