Step 1: Picking out a sweater to take apart
There are certain qualities in a sweater that you have to look for when you want to take it apart (to make it into a sweater or anything else.) First and foremost: The seams. This has to do with the fundamentals of knitting. The most important rule is stay away from serged seams. Most of the time, when a sweater is constructed in pieces, the pieces are just sewed together. When this is the case, all you have to do is cut the pieces apart, and unravel the pieces. This way, you can (theoretically) unravel each piece of the sweater into one long piece of yarn. If you imagine a sweater (or any piece of knitted fabric) to be one piece of yarn, weaved back and forth, the process looks something like this barbaric paint drawing I made:
A serged seam is a kind of seam, made by a serger, that cuts the fabric while sewing it together. When this occurs, if you look at my crude drawing, it's pretty easy to see why this could be a problem. At each edge, the yarn is cut apart. Now you have a billion little fragments of yarn that are pretty useless.
I'll upload some pictures of serged/not-surged seams with my next sweater project, but a simple Google search can be a big help.
Another basic rule-of-thumb that I follow is don't pick a sweater made up of yarn too light. Once a sweater is worn and washed for years, the yarn gets significantly weaker. You'll have to spend hours upon hours unraveling a sweater made up of fingerling-weight yarn, and I'll bet that it will break a lot in the process. It's a pain in the butt, so I do not recommend it.
Step 2: Unraveling
The first thing you have to do to take apart a sweater is take it apart into it's components. Generally, a sweater is sewn together using thread that is a little different from the yarn the sweater is actually made up of. Identify this yarn, make a few snips and pull the fabric apart to see if it worked. If you can see the space between the pieces, you've found the seam thread, keep going. Cutting the knitted yarn a couple times is not the end of the world.
Unravel the piece into a skein, not a ball right away. There's one additional step that I recommend.
After the piece is in a skein, it's time to wash it. Once you've gotten to this point, you probably think this yarn is too lumpy to work with. Here's a picture of a piece of a sweater unraveled into a skein before washing:
Take the skein, tied together at a few points, and put it in a sink filled with water and soap. You can use regular dish soap, or crazy-people soap like I did.
Swish it around a little bit, making sure all the yarn got soapy. Drain the soapy water and rinse the yarn thoroughly. Hang to dry! Here's what the same skein looked like after washing & drying:
Disclaimer: I love buying new yarn. Recycled yarn definitely isn't as nice as brand new right-off-the-alpaca yarn. But if I bought nice new yarn for every project I do, I would not have enough money to eat. Literally. I usually buy nice yarn when I'm knitting something for someone else as a gift. If you look at my Ravelry, you'll see that the mittens I made for my grandmother for mothers' day was new yarn, as well as the mittens I made for my sister for her birthday. I'll post some pictures of the sweater I made on Ravelry if you'd like to see the finished product. Stay tuned for a full step-by-step sweater-to-sweater conversion soon!