Last night I cut apart a cotton (knitted) sweater, planning to make it into a bag. Previously when I did this, I unraveled the entire sweater, however this sweater had lovely diamonds knitted into it and I didn't want to destroy them. So, I cut the sweater at the arms, so I had a straight piece of knitted fabric. Then I sewed a seam at the side and unraveled one arm's worth of yarn. The ribbing end of the piece(formerly at the bottom of the sweater) will now become the top of my bag. I attached my yarn at the raw edge of my bag and picked up the free loops; I began to knit, to make my bag deeper. That is where I'm at now-see picture.
Next, I plan to begin decreases to make an oval bottom to the bag. After that I will knit an eyelet row above the ribbing, and knit a strap. Then the bag will be finished, and it will have cost me just a couple of dollars.
A word of warning about purchasing used sweaters to unravel: many sweaters these days are not knit per se. The pieces are cut from large pieces of knit fabric, much as you would cut the pieces for a shirt from cloth fabric. You DON'T want to use these sweaters for unraveling as all you will get is one yard long pieces of yarn! (Ask me how I know!) You want to buy a hand knit sweater; these sweaters will be sewn together with the yarn they were knitted from, and they will have very finished-looking seams, unlike the seams from the former sweaters which will have seams with little short pieces of yarn kind of poking out. You may get some of the wrong sweaters at first, but you can still use the yarn from them to crochet little amigurumi toys, etc. You don't need to waste it. Good luck and happy hunting and knitting!