This is my very first Guide I'm writing.
I saw Galio in Elemtz tierlist to be "Hard to place on a team" and wondered why he would place him there, since he is not, as many say, a "bad Amumu". He has a lot of Utility and can turn fights to your Favor.
Allthough I admit he needs some Buffs, he shouldn't be named in one breath with Eve.

Q and E: Great for making your team catch up or engage with your Ultimate, also great for kiting

W: Very Helpfull against DoTs. Increases your AP through your Ultimate if used. Great against GP
if he harrases you with his Parrrley ( You will get 2 damage each tick, but your Shield will
Heal you for much More.

Flash: It is a must for Galio. It helps you place your Ultimate.

Exhaust: After your ultimate it's most likely that an enemy will Flash away to get a better
position. If you Exhaust him it is easy to catch up for the Team. It is also a great
Supportive skill for a tank, because it helps him to protect his carry.

I'm taking basic Tank Masteries, allthough I prefer to take 2 Points in "Defensive Mastery" for the laning phase.

The Runes give you a lot of Magic resistance in a proceeding Game. Since you need around 220 Mres/Armor with a Tank and Galio lacks damage if he goes for pure tank, you will need these Runes.

Movement Speed Quints together with your Righteous Gust will Help you to catch up to enemies, so that you don't HAVE to use Flash to engage.

Since you will most likely go to the top lane you want to stay for a long time, but also pressure the enemy that's why skill Q and W first and since the Lane isn't "long enough" for your E to have a huge effect in catching up, you don't need to skill it early.

If you start off with a Null-magic Manteu you have around 35 AP which gives you a rather big damage advantage to your enemy. If however you think your enemy will be a DPS-char that does not have a DoT, then you should go for Dorans shield.

The Chalice of Harmony is a must. Without it you won't be able to stay on line for too long. You will sell it later, because the Mana you get from BV is enough for teamfights.

Next is Sunfire Cape. It is easy with Galio to stand in the Mid of the enemies Team, cause of his Movement speed buff and his Ultimate, so you will deal a lot of Damage with it.

I prefer taking BV over FoN. This is because spells that enhance Auto-attacks still work on you if you use your ultimate and can cancel it. It also helps to counter other tanks CC's such as Amumu's, that way your team and the enemies will be disabled and not only yours.
You also don't get enough HP with this build if you build FoN to make it worth buying it.

Next is Thornmail. It will hurt carries that attack you during you ultimate, which is very Important.

Now for Abysall Scepter. it gives around 100 AP with Galio and also increases your damage output greatly, because of its Aura. It also gives the missing Mres for you to reach a value higher than 200.

If you can Purchase a Deathcap in the end you will have lots of AP and you will be able to make your Ultimate have a great impact on the game.

You will most likley be on the Top lane, because mid is AP, bot is carry+Support and you will have a jungler.

If your enemy harrases you with a DoT, you can easily heal yourself with your Bulwark. Since it heals you with every tick of the DoT and the ticks don't deal a lot of damage in the Beginning.

Espacially GP is an easy laning enemy for you. He will most likely harras you with his Parrrley, which adds his Passive on you. It deals 2 Damage per tick using Bulwark you can easily recover from the rather Hard hit of GP's Q.

If your enemy used all of his spells you may also attack him to get the Aggro of the enmies Creeps in order to Heal you through Bulwark.

Be careful not to Farm with your Q, it will give your Minions a huge Push and if the enemy is intelligent he will deny XP and Gold from you.
However if you harras the enemy correctly he will have a hard time to recover, because your Q hits very hard in the beginning.

Ganking with a jungler is very easy with Galio. Most of your enemies will understimate you, because Galio is seen as "weak" or even "useless". So if you engage the fight without the jungler near you, they think they have a high chance of winning against you. Your jungler needs to attack when you hit a Resolute Smite.

If you proceed like this it will easy to secure your line.

You most likely will have enough farm until now to help out other lanes.
Ganking Mid with Galio is rather Hard so you should get the Bot lane.
Try ganking from the last way you get on the lane if you are on top of the screen, if you are on the bottom screen you will have to gank from the bush.

In teamfights you want to protect the carry. So before you start your ulti make sure to give him your shield if anyone counterflashes you. If somone counterflashed or a strongly feeded enemy survives after your Ultimate Exhaust them so they don't kill your Carry.
After your Ultimate is ended directly dish out some damage with your Resolute Smite.
If your team needs to flee set up your E to get a MS advantage. Do the same if you need to chase, but only do that if you are able to. Otherwise protect your carry with a well set Slow.

Farming with Galio is really easy just use Q and E to kill creep spawns

If you follow this Build your tank/Damage stats will look like this:

Health:2765
Armor: 243
Magic Resist: 222
Ability Power: 434
Magic Penetration: 20
Movement: 397

I'm rewriting this because on top Rabadons and Ardor are not calculated and Galios passive is also missing.

Your skills will have the following effects:

Runic Skin: Ads 111 AP
Resolute Smite: Deals 603 Damage
Bulwark: Heals for 215/172/129/86/43 This means if you target yourself with the Shield, all
damage that deals damage below this Values will deal no damage to you
Righteuos Gust: Deals 457 Damage
Idol of Durand: Deals 700- 980 Damage