Galio counters most ap champions on mid, thanks to his passive Runic Skin, 50% of his mr is converted into ap. So even if you don't buy as much as normally ability power, by playing ap champs, u still have enough dmg to make your opponents feel pain. As starter I usually buy Null-Magic Mantle + 2x hp potion or 2x Faerie Charm + 3x hp potion or boots + 3x hp potion. It depends how i feel against opponent.
As first item i rush Chalice of harmony, without this item Galio is not so scary. It regenerate mana a lot, what is very important, for good lane control. Also always I buy Mercury's Treads and Catalyst the protector. In early game it's important to always have mana, you can poke and in case of gank escape by using your e (for incrase movement speed). Always remember to shield yourself when fighting, cause it's healing a lot and decrease dmg wich you will get.
For full build I usually build Mercury's Treads, Athene's Unholy Grail, Banshee's Veil, Rabadon's Deathcap, Abyssal Scepter, Void Staff/Runic Bulwark/Warmog's Armor (depends on the commposition of enemy team).

When playing against ad champions on mid like Kha'zix (with probably ad top also) your passive is pretty useless. So all u need is armor and ability power. Good idea is to buy Seeker's Armguard. I usually start with cloth armor and 4x hp potion + mana potion. Ninja Tabi, Banshee's Veil im my opinion - must have. Abyssal Scepter might be helpful, it decrease mr of your opponents, grants you mr and ap. Good idea is to buy Frozen Heart or Randuin's Omen for slow and even more cc. End with Rabadon's Deathcap or Zhonya's Hourglass. When there I don't usually buy Deathfire Grasp but against tanky team it might work. When through the game there is a lot of teamfights, you can invest in Locket of the Iron Solari or Sunfire Cape.

Farming while playing Galio is easy. Use your q for killing range minions, then if they are still alive finish them with e. When fighting remember to shield yourself for more defensive and good heal. Remember to ask jungler for blue buff, sometimes they forget to give it to you. With blue buff u have almost infinitive poke in connection with your damage make your opponent mad. You can easy out farm him. Your e skill will help you to quicker roam. Flash + ultimate with good timing = easy kill. Don't be afraid to shield allies in team fights,or when doing baron/dragon you will also receive heal.