I have tried various coupon storing/organizing methods over the years. There are advantages and disadvantages to each. A binder makes it easy to see each coupon, but folding them all to fit in the little pockets is a real pain. A box is nice but cumbersome. What it boils down to is that you will have to find out what works best for you. And just because you start out using one method doesn't mean you have to stick with it forever. If something isn't working for you, try a new way!
I use a large accordion style check file from Walmart (costs about $4) for my coupon box. It is a bit longer than a regular coupon organizer but I like that I don't have to fold some of the longer coupons. It also has a little compartment in the front that I use to store scissors, a pen, and a calculator. Put your name and phone number on your coupon organizer somewhere so if you lose it then you'll hopefully get it back. (Ask me about how I learned this lesson!)
Organize your coupons so you can find what you need quickly. I organize my coupons by category. I use envelopes to divide each category. For example, in my "mouth" section I have envelopes for toothpaste, mouthwash, toothbrushes, floss, etc. I also have one envelope just for "free item" coupons so they don't get lost in the mix and expire on me.
On some of my envelopes, I jotted down the known lowest price (like from Sams or Aldi) for items I buy frequently. That way I know if I can get a better deal with the coupon then I should stock up. It's like a price book built right into my coupon file.
This is probably obvious, but don't forget to go through your coupons every so often and remove the expired ones. I hate that... I think I've found a great deal and then notice the coupon is expired.
In the back of my coupon organizer I have an envelope for each store I shop at (Piggly Wiggly, CVS, Walgreens, etc) and as I make my list I stick the coupons in the store envelope. Do bring your coupon organizer along though even if you pulled all the coupons out ahead of time to match your list... you never know when you'll find a deal. If you don't want to bring it in just leave it in the car.
I, personally, don't cut out every single coupon, although I know lots of people who do. I just cut the ones I know I'll use and save the rest of the coupon "inserts" in a file folder sorted by date. When a deal blog or someone mentions a coupon they usually tell you which insert it was in and you can go get it out when you need it later. You can mark the date on the inserts or if you look at the fine print on the spine of the coupon insert you will see the date the insert was issued.
You can also search MoneySavingMom's "Coupon Database" to see if/when a certain coupon came out. This has been especially helpful since I started shopping at CVS. I have bought items there that I normally would not have saved the coupon for, but since it is free, I'll get it.
This method has saved me a bunch of time... and space in my coupon file! The down side to this is if you don't have a coupon with you that would get you a great deal, like if you find something on clearance for instance. But it is still much better than cutting out the ones you think you will use and tossing the rest!
Last year I won this bag and it has now become my coupon bag. I just cut the top off my accordion file and put it in the bag. The bag has generous side pockets for various necessities. It fits perfectly in the front part of the cart and, best of all, it's cute!