In truth, you can fold origami out of most types of paper, but traditional origami paper is the easiest to use in my opinion. Origami paper is wonderfully thin paper that comes cut into squares and ready to fold. Over the years I have purchased paper from a variety of websites, but my favorite place of all to order paper from is Amazon, and I've included links here so you can find them more easily. Their prices are quite reasonable, their variety of colors and sizes is awesome, and their shipping is usually pretty fast.
I love these solid colored packs of Toyo Paper that come with 100 sheets. What's nice is that they come in so many different colors, and you can order just the colors you need and use a lot of. They are colored on one side and white on the other side. These papers are thin traditional origami papers.
If you're looking for a pack of assorted colors, then I highly recommend either the Aitoh or Toyo brands. Their colors are vibrant and I've worked extensively with both. Michaels used to carry Aitoh, so it was my go to brand when I needed paper urgently, but they don't carry origami paper anymore (another reason I use Amazon now). Both of these brands have variety packs with lot of colors.
When it comes to handmade Washi or Chiyogami Papers, I prefer to get them by the sheet (24"x18") from The Paper Source or online stores that specialize in these papers. These papers can be rather expensive and I find that getting them by the sheet and cutting them myself saves me a lot of money. I usually order these large sheets from: The Paper Source, Mullburry Paper, Kim's Crane or Paper Jade.
If you're just looking for any random assorted pack of these handmade papers you can get them on Amazon - just be aware that some packs will have plain sheets mixed in with the prints, so read the descriptions and feedback. These that I've linked below only have printed sheets in them.