Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
This is a rough draft of my Galio Build. I will gradually add more details but this guild is mainly for my own purposes to remember my build.
When laning with a partner, you should get Bulwark (W) at lvl 3 in case they, or you, get harassed and eventually dived because it might just save them.
Instead of rushing GA, it is better to get Banshee's Veil for the extra HP, Mana, and MR (=more AP)
If it is a long game, sell your Chalice and get the extra AP of Zhonya's Ring with its great passive.
The Glyphs and Seals are experimental, it could be swapped with flat cooldown for more harassing but the MR adds to his nuking power.
The extra EXP gain from masteries can be used to nuke and get your first kill once you get lvl 6 before them. After harassing them, let them push towards your tower, aim for the squishy and W yourself, flash in, ult, and nuke with your E-Q combo then ignite as you chase them down if they are not already dead.