My name is generalgarchomp i've been a galio main since the ap rework and played him weeks before that. I love how he can burst people down in seconds and still have insane map pressure with the ability to go to a lane and instant snowball it, then go back to lane with a large gold advantage like it's no big deal. In my opinion a good galio can easily carry games even in low elo DUE to this pressure, he can keep his allies safe while getting fed himself or when he ults bot to save from the 2v3 gives his adc (that had been losing lane) a triple thus getting them snowballing back into the game. This build is based solely on stupid burst with your combo and SOME hp. You'll more often then not build defensive items unless they have a lot of the dmg type or one of them is fed and you're doing meh.

Also a little known fact (i think) if you start your w before cc once you get cc'd it will automatically go the full length. I've seen this most against roots and zoe's sleep so it could just be that.

The page i included is the one i use, however you can switch it for aftershock if you feel you're too squishy, the runes in that case will be the ones on my bruiser build. A lot of people take comet but i don't see why as much. Galio's more of a up close and personal combo champ then a poke based champ so comet is more situational imo, plus the domination tree is nasty with him.

You max q first because you want its scaling to be capped quickly. E second cause its damadge is insane due to high base dmg and 90% ap scaling and is your main combo starter. And w last because it's not as effective on ap as it was on tank and is now only really used as an engage, escape, and to tank things you can't dodge and might not live otherwise.

There are 2 main ones, the normal one you'll use almost all the time is e into empowered auto q then w and another auto. However sometimes you'll need to engage from a distance whether it be to set up for a gank or on the enemy team, in this case you charge w and flash taunt, then you q into e empowered aa. the combos are made to keep them cc'd as long as possible in the q so as to Maximilian the damage.

As said your combo is as follows e to engage and knockup, auto for phat damage and lich bane proc,q for moer phat damage, and taunt to keep them from escaping.
the damadge on them are
e-100 / 130 / 160 / 190 / 220 (+ 90% AP),

empowered auto-12 - 80 (based on level) (+ 100% AD) (+ 50% AP) (+ 40% bonus magic resistance),

lichbane-75% base AD (+ 50% AP),

and lastly q-
gusts(each gust seperate)-50 / 80 / 110 / 140 / 170 (+ 80% AP)
tornado half second damadge-10 / 13.3 / 16.7 / 20 / 23.3 (+ 20% AP) (+ 1% per 100 AP of target's maximum health),
and lastly the full tornado damage(this is most important part because it is waht will be hitting the most)=30 / 40 / 50 / 60 / 70 (+ 60% AP) (+ 3% per 100 AP of target's maximum health)

Add to this the dmg from electrocute- 50-220 + (50% ad, 30% ap) it may not seem like much but TRUST ME it makes a difference.

The reason i put these is mainly to show the SHEER damage of this combo... and this is sans ult but for sake of the rest i put and my ocd the damage is as below
150 / 250 / 350 (+ 70% AP)

Your ult is an insane amount of map pressure and can be used to snowball a lead for you and others, it's the main reason galio is played mid even as a tank. As such you have to be aware of two things, enemy and teammates positions in their lanes, and ally health bars. If their getting bursted to nothing then ulting won't save them. Let's say bot's about to get ganked and are in the middle of the lane, you path down and ult the one in the middle of the enemies, usually your adc, this often results in 3 kills on your side and often no deaths, a gank that woulda given enemies a double maybe gave you or the adc a triple. Same for top, they could be about to be ganked or die 1v1 then you ult in and save them, putting them ahead or keeping them even. The strength of it is likely the main reason he's banned in pro play.

You have 2 main engages flash taunt, and knockup. flash taunt is better for team fights as it is the only real aoe engage you have. e is for when there are only a few or someone chasing a team mate, that or someone using a cancel-able ult. But what he's easily the best at is follow up. whether it be using e on someone cc'd, on someone an ally ww ulted, or even better ulting whoever ran headfirst into enemy lines. Your ult can save lives sooooo easily, whether they got caught out by or is about to be barely killed by an enemy adc. For example, one time my ally kog ma was fighting an ashe they were equal matched in damage but she out lifestole him, i ulted in letting him take less damage where he healed more and got the kill, moreover immediately after he was hit my the enemy jg ez (forget if it was q or ult), he survived with less then 100 hp and escaped. The dmg reduction from my ult saved him from taking enough damage to be finished off.
The biggest problem is knowing when you can kill them and when you can't escape after ulting and hopefully saving whoever you ulted. And when the ults will save your ally or be pointless and get you killed too.

Your waveclear is amazing empowered auto and q, q first and kill with ao on empowered auto, and e on minions you fear you'll miss. late game take raptors when your jg lets you and steal enemy wolves and raptors when you can.