I sell the low quality Alchemy material. The "Mysterious Bag" EX skill allows me to not worry about running out of inventory space.

There is no way to improve purity, but you can combine them to get certain other items, some of which can come in handy (like the one that shows the locations of placed encounters on the map). Quality is irrelevant for most of them.

Yeah, either sell or combine low purity stuff. I mean you can also just use them on weapons and armor in the beginning since you'll replace all your equipment eventually anyway.