There's a reason grocery stores use plastic bags; they're inexpensive. Paper bags are expensive enough that they are not used in a majority of stores. What does this mean? It means that if a grocery store can no longer use the inexpensive option, they must resort to a more expensive one. That means that prices will rise and wages will lower in order to meet their bottom line.
The benefit of this law is supposed to be less pollution; less plastic pollution specifically. I expect that it will meet this goal, but the cost to the consumers will be substantial enough that the plebeians in California will demand price controls on the goods from grocery stores. Ultimately, I think this is a bad move for California.
But I'm biased.