Here's how I have my 12x12 cardstock stored. It is in vertical storage boxes that I bought from Oriental Trading Company. They are very sturdy and hold LOTS of cardstock. I was having a problem with trying to find my different patterned cardstocks from TAC. So I made separators from a piece of plain 12x12 cardstock. Then I added a strip of white folded in half to make a tab. On that tab, I put a label listing the paper's name and then all pieces of that paper go behind that tab. It has made it much easier for me to find a particular pattern.
Also, you can't see it very well in this picture, but I sorted my 12x12 plain cardstock in rainbow order. All reds together, all oranges, all yellows, etc.
Here's the 8 1/2 x 11 cardstock. I used a similar idea of vertical storage and sorting by ROYGBIV order with this cardstock. The containers are plastic magazine holders from Walmart. I turned them on their side so that the opening was facing the front of the shelf on which they are sitting. The cardstock is then sorted by color with a separate holder for all my neutral colors. I had a hard time seeing when I was running low of a certain color so I took a piece of 12x12 white cardstock and trimmed it down to use as a divider between each color -- now it's very easy to see which colors are running out.
Finally, here's my 8x8 patterned paper pads. I have ALOT of these. They are all sitting in a plastic tub that I got from Walmart that I decorated. I can easily flip through and find the pattern that I want this way.