Great Ball at PokeStops

At level 12 From This Question you start to obtain great balls when leveling.

I am currently level 6 and have only ever gotten normal pokeballs from pokestops.

Is there at any point (possibly after 12) where pokestops start giving better balls?

I am level 12 and once you "unlock" the great ball it will be dropped at pokestops. They are harder to get than normal pokeballs. This goes for nearly every item (eg. razzberries, super potions, etc).

All items that will be given at pokéstops require you to unlock them first.

After you reach the required level then you will start to receive them as well.

Yes, you receive Ultra Balls at level 20. Note that once you unlock an item, it becomes obtainable at PokeStops.

Source: http://www.ign.com/wikis/pokemon-go/Level_Rewards_and_Unlockable_Items