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?
Yes, you receive Ultra Balls at level 20. Note that once you unlock an item, it becomes obtainable at PokeStops.
Yes! We unlock a Pokemon Go great ball at level 12 and an ultra ball at level 20. But I'm really surprised that everyone forgot Master Ball! I was expecting master ball as well in Pokemon Go!