Rather than clipping out every coupon, you can just keep your Sunday paper coupon inserts whole. Take a pen (I like a Sharpie marker for this) and write the date the coupons came out on the right hand top corner of the insert.
Subscribe to the CouponMom.com website. Each week, check out "Deals By State." The CouponMom matches up the coupons to the products that are on sale, and even figures out the final cost after the coupon for you. Use the drop down menu to choose your state and then your store.
She uses a code like SS 6-15 next to the item, which means that the coupon was in the SmartSource coupon insert on June 15th. I have marked my inserts by date and plopped them in a drawer. I just pull out the matching insert, flip through until I find the coupon I want to use, then cut it out. Ignore most of the other links on the Coupon Mom website, because there is so much going on in that website. Much of it is just affiliate marketing/sales links, and you can easily get distracted and off track.
Another way I do it is to cut out the coupons from the inserts when I first get them, but only the coupons I know I will use. Then I mark the insert with the date and stick it in the drawer.
Publix puts out their new sales circular on Thursdays. Kroger's (and most other stores') sales ads come out on Sundays. Thursday or Friday is a great day to shop because is you compare the store circulars, you will know which store has the best deals on the items you are most likely to buy. If you are really intrepid, you could have a list for each store and hit them both for their best deals on the same trip. If you do this, I recommend you put a cooler in your car, as it will allow you more shopping time without worrying about the cold/frozen items in your car. Using a cooler also can make it easier to bring everything into the house and speeds up the putting away time.