The integral Chow groups are infinite dimensional. Can we say something about their dimension, for example, how many elements are required to generate them? Or their vector space dimension after tensoring with $\mathbb{Q}$. What is known in this direction? I recall hearing some statements whose proof uses the theory of Chow varieties.

Also, is there a good reference for Chow varieties. I am looking for something which will assume familiarity with algebraic geometry at the level of Hartshorne and which uses that language. Since I am reading Vistoli's notes on stacks, reference to some construction using the language of stacks might also be helpful.

EDIT: Removed the remarks on countable generation.