The canvas I use is the same for both methods. You can buy cotton or linen canvas by the yard at an art supply store or you can get it online at one of the art supply houses. There are a lot of choices, you can get it primed or unprimed. I buy the primed linen canvas. At one point I bought the unprimed canvas, stretched it myself over stetcher bars and then coated it with rabbit glue and oil primer. What a lot of trouble and mess that was. I decided I needed to spend my time painting, not stretching canvas.
The photo above is of some primed linen canvas. You can see the natural unprimed side and the primed white side. You just cut it to the size you want with ordinary kitchen scissors. I usually use a straight edge and a pencil to make straight marks before I start cutting. The linen canvas isn't cheap, I think it costs something like $45 a yard, but it's 54" wide.
Okay, now that you've got your nice cut piece of canvas, what to do with it? You could stretch it on canvas or glue it to a support and paint it later, OR you could tape it to some foamcore type material and paint it first. That's what I've been doing lately. That way if I don't like my painting, it's no big loss. Below is a rectangular piece of canvas taped to some white foamcore. Foamcore also comes in black. You could conceivably use anything that is rigid, a couple of days ago I used a big piece of cardboard for the painting I did in art class. You can see how I use this method of painting on unstretched canvas here on a post I did a couple of days ago. :
I learned about cutting canvas and taping it to foamcore from Lesley Rich. She cuts a bunch of canvases in various sizes and takes them along when she travels. There isn't much weight involved and they don't take up much room. Once the paintings are dry, you can just stack them up and they hardly take up any room at all in a suitcase.
If you plan to paint a bunch of 6"x6" paintings, you'll have to cut your canvas 6 1/2" x 6 1/2" to allow for that little 1/4" that gets hidden behind the frame. I cover as little of the canvas as possible with tape, that way I can paint almost to the edge of the canvas.
If you plan to stretch your canvas, you need to leave a lot more canvas around your painting. A 6"x6" painting will probably need to be on about a 12"x12" piece of canvas. You'll understand as you read more here.
When I'm finished with my painting, it's dry, and I like it well enough that I didn't throw it in the trash, then it's time to get it ready to frame. At this point I can make it just about any size I want by cutting off the parts I don't want IF I'm planning on glueing it to a support. Just remember to leave that 1/4" that'll be covered by the edge of the frame. I use YES! paste, which I bought at the local art supply store, and I use some kind of brown masonite-type material I bought at ACE Hardware. I think you could use foamcore or similar materials but you probably want them to be archival or acid-free. One disadvantage to using such a rigid support is that you have to have a way to cut it (a saw) or you could probably take it to a framer and have him cut it, or ask a carpenter. I just use a brush and put the glue on the backside of my painting, then stick it on the support. This YES! paste is very thick though and I had to add some water to get it thin enough to spread. There are other pastes that will work too, this is just the one the art store had.
After the canvas is glued to the support, I would wait a few days and then varnish the painting, then take it to the framer when the varnish is dry.
The other way, which to me is more complicated, is to stretch the canvas across stretcher bars. You can buy the stretcher bars at an art store for around $1 each. You have to put them together at the corners, they just have notches that fit together, BUT you have to keep those corners at 90 degree angles. You need two pieces of equipment, the "grabber" thing to really grab the canvas and pull it around the backside of the stretcher bars. Then, while stretching the canvas taught, you'll need to use the staple gun to secure the canvas to the back of the stretcher bars. At our local art store there are stacks of xeroxed instructions on how to do this. Below is a photo of the grabber thing and my heavy duty staple gun.
Once you have your canvas stretched, and the corners are all squared up, then you varnish the painting, let that dry, and take it to the framer.
For me there are two reasons to use unstretched canvas: 1. I can cut a bunch of rectagular sizes and if I don't like what I did, it's easy to throw away. Great for small sudies. 2. It gives me some flexibility with the composition. If I decide I have too much sky on a landscape, it's easy to get stretcher bars a little shorter or cut off that part and glue it to a smaller support.
I think stretching canvas is one big pain and I would much rather glue my canvas to a support, but that's me. Most of the time I paint on prestreched canvas or canvas already glued to a support that I buy on-line from Tim at http://www.newtraditionsartpanels.com/.
I hope I've explained this well enough. I'll try to answer any questions though I'm sure there are lots of ways to do this.