OK this is just how i personally play Galio. I have seen some good AP Galios and some OK Tank/AP, but after this past patch; i find him to be a much better tank. You can obviously change the items and tweak certain Runes/Masteries to how you see fit, but again this what i like.

Galios skill set can be very useful to a team, allowing you to save team mates, initiate team fights and still do damage.

Resolute Smite:
This move can be deadly late game and a good harass early game. The damage for this move as a tank is very good, plus it has AoE damage. But the key to this skill is the slow it has on your enemies. By end game it is slowing your enemies by 40% for 2.5sec! This can means death for any enemy champion that is trying to flee and you can slow allowing one of your carries to catch up and kill him.

Bulwark:
This move is can be used a couple different ways. 1) you can use it on yourself before ulting to allow your self to start with some bonus armor/Magic Rest. 2) Shield a carry champ right before the go into battle to allow you to regain some life back while still fighting and letting them get some bonus armor/magic rest.

Righteous Gust:
This is a does decent damage and can be a very strong support move for getting away from the enemy or catching up to the enemy.

Idol Of Durand:
This is Galio's signature move. Giving Galio bonus armor and magic rest. While he channels this move he cuts all damage in half and taunts all enemies within its radius. Then after it is done channeling 5% of all the damage dealt is redistributed to the enemies.
Now for the most part when the enemy is stuck in Galio's ult. the first thing that they think about is "GET OUT!" Here is the thing about leaving Galios ult. if 4 people start in his ult and then 2 get away all the damage dealt from ALL 4 CHAMPS are re-dealt to the 2 remaining.

Start/Early Game:
When the game starts Galio doesn't have a lot of mana and his moves early game use a lot of his mana very quickly. So i like to start off by grabbing a Meki Pendant; and then 2 health pots incase you loose a significant amount of life. Although, taking Meki Pendant leaves you rather squishy, take this chance to hang back and harass with Resolute Smite and Righteous Gust.

Mid Game:
i consider mid games to be levels 6 to 14 for Galio. During this time, if everything is going well, you should be making Galio more tanky. During these levels you should be purchasing Mercury Treads, Warmogs, Banshees Veil, and Sunfire Cape.
Galio can also start playing more aggressive, he will have substantially more life so he can take a view hits.

Late Game: If you can successively get all your items or even most of them by level 18 you will be a very good asset to your team. You can initiate team fights and then last through the fight to help your whole team. 1 very useful skill combo i like to use late game is using Bulwark on myself flashing in to the enemy team while they are huddled together and then using my ult. This will allow your team to jump in and have the enemy team start fighting with some life missing.