To be fair, since Galio was nerfed a lot of people started to think he sucked, in reality however he doesn't.

This peculiar build is quite effective on this character, it let's you main tank/off-tank quite well because of the very high armor/MR, that along with about 3k HP, Galio's Bulwark, Banshee's and Guardian Angel can make your hero extremely tough to kill.

I prefer to go off-tank and get the enemy by surprise with Idol of Durand, but it is not impossible to get them while being the only tank, if your team supports you.

With this build Galio can easily solo lane or middle against pretty much anyone if he plays well due to his high damage output and survivability, 2 high range (Kog'maw, MF , Lux , Caitlyn and whoever else) can be troublesome alone but not impossible if you time your Resolute Smite well, along with your Righteous Gust.

For the runes and masteries, they are specially related to Galio's forte which is dealing damage to big groups of enemies while holding them.
This build let's you build up a Galio with the possibility of having between 50 and 75 AP at level 1 depending on which item you pick up first, I'll let you imagine how hurtful this is. Let alone having a basic 15% Magick Pen from Archaic Knowledge and and 10 more from runes.
I go for Magick Pen and AP runes for this reason. The same goes for Magic resist rune, not only do they aid with defense but used along with Galio's passive which converts MR into AP, it's great. 8.5MPen, 24AP and 6.7MR (3.4AP bonus for Galio)
For the masteries, I go for 9-21-0 but in the defense I avoid Dodge, simply because the more hits you get during Idol of Durand , the more it hurts and dodging is bad.
Not much more to say, you get the point.

As for items, this is where the debate will come alive I guess. I believe Galio is pretty much built for these items.

First off, Galio's main problem since the nerf patch is mana. His spells cost A LOT of it early on, which is why I go for the Mana Manipulator even if I play with clarity. There is a few reasons for this item over others. First, even without a Doran's Ring or an Amplifying tome, you should have about 50ap at level 1 which hurts, a lot, so having more is not really necessary unless you are solo laning.

Another good point is, if you play with another mana hungry hero, he'll benefit from your choice as well.

Finally the main reason behind it is that if you play well, Galio has so much range that he shouldn't be hit by anyone, and even if he does, tanking masteries we're chosen for this.

The second item is the Merc boots, hands down, They give MR, which then conerts for AP and also a 35% debuff reduction, by far the best boots for a casting tank.

The third/fourth item depends on your enemies and how the game is going, if you encounter a lot of mages, Banshee's is the way to go, HP/Mana/MR and spell block, if you are facing a lot of ADs however, Thornmail is the best choice, since it makes them pretty much obsolete because of the 100 Armor and 30% magick damage thorn effect. If you are facing an evenly split team, consider finishing your Soul Shroud for the high HP boost and CD Reduction, which helps a lot. Getting Sunfire over Thornmail is advisable if the enemy has only 1 AD dps, otherwise I find Thornmail to be way more effective with Idol of Durand.

As for the fifth, I pretty much always go for Guardian Angel, since Idol of Durand might get you killed and focused, ressurection is very very useful for Galio, not mentioning a bit of MR to boost his AP.

The last item, in my opinion is a personal choice, I like to go for Abyssal Scepter for the very nice 70AP and 57MR (about 26AP for Galio) and the 20 MR debuff which can be nice as well. If you are having trouble surviving however, a Sunfire Cape, a Force of the Nature or a Rylai's can be nice options.

Not much to say, get Bulwark only once at low lvl for the MR boost to hit with then focus with dmg dealing abilities until your ult. The shield is great, but not THAT much.

Timing Resolute Smite and Righteous Gust on an enemy can have devastating effects early on, not to mention that by level 3-4 you can 2-shot all of the caster minions which is great for income in 2 spells AND still stay very safe from a distance. If you have spare mana, shielding yourself before attacking is a great idea for the MR boost which means AP boost.

Resolute Smite slowing effect combined to Righteous Gust slow/haste effect is a very effective method of hunting down enemies. Combined with Ghost/Flash it's very nice.

Idol of Durand is very strong ultimate, especially against heroes who cannot silence/stun you, not only does it taunt but it lets you dish out very high damage on multiple enemies if timed correctly.

I go for Clarity and Ghost/Flash

Clarity is obviously to counter Galio's lack of mana, it is pretty much the most needed summoner spell for this build.

Flash/Ghost depends on the situation, if you are main tanking, go for Ghost, it helps charging in the enemy upfront to ensure being their target. If you are off-tank/ganker, go for flash, if your tank is in the fray, flash through a wall or from an unseen spot to land your Idol of Durand for an almost assured good result. Flash is also good for main tanks wishing to tower dive, but this often means suicide unless you have Guardian Angel already but 1 death for 2+ death is a decent trade in some games, no?

I'd say the best choice other than those is really Fortify, because yeah, Galio can AE taunt, which means that when 3-4 enemies are attacking a tower, Flash/Idol/Fort combo is DEVASTATING for the enemies. Even more so with the fort mastery which makes the tower do Splash Damage. This tactic works VERY well will a coordinated team jumping in right after.

I see a lot of Galio's with Teleport and Heal, while not the worst choices, they aren't the best either, Teleport becomes a bit obsolete as the game goes on since with Ghost/Righteous Gust you can effectively run around the map with no trouble. Heal is also a bleh case, Galio doesn't take that much dmg past lvl 4 due to high resistance and decent hp so unless you lane with a low hp buddy it's not really the best choice.

Exhaust is somewhat obsolete for Galio, you WANT them to hit you for Idol and Galio already has ''hunt the enemy down'' mechanics with his Smite/Gust combo.

Revive/Smite/Rally are no good.

Pros
High tanking capabilities, even better off-tank capabilities
Terribly tough to hold back, which means easier Idol hits.
VERY high damage and harassing capabilities during most of the game due to high range/damage AP based abilities.
Very high mobility even without Ghost/flash due to Righteous Gust.
By the end of the game, 200-250 AP with a good magick penetration adds up nicely to the team damage output, not mentioning his Idol of Durand using Thornmail/Sunfire.
Incredible Farmer due to high damage abilities early on.
High damage scales with magick resist which means dmg go up at the same time as survivability.
Can solo lane effectively with his Bulwark HP regen and Clarity.
One of the best team based hero there is.
Will utterly DESTROY casters and pretty much laugh at melee's end game.

Cons
Still hard to maintain good mana levels even with clarity mid-game, try to aim for the blue buff.
Not as much hp as most tanks during most of the game which is why he's better used as an off-tank.

All in all, an underestimated hero after the patch that can be very good if well played.

Give this build a try and see for yourselves.