Don't use any stretch fabric that would be best for swimsuits, the stuff just doesn't hold up. Ditto for polar fleece, it's got too much stretch in it, for my taste.
If you go with cottons or cotton blends, they'll take wear and tear the longest and the fabric doesn't snag like nylon or rayon. And if you're using sewing machines, it's always easier to sew seams on fabric that isn't trying to slide apart (some polyester is really terrible about this). But you CAN make bags out of most fabrics.
I think someone posted a site where it showed the steps to making a round bottomed drawstring pouch, a while back (December?), and I know there's sewing sites that carry info on it too. The hardest part is actually stringing the cording through because one string goes around to the right, the other string goes around to the left (so both cords are in the same channel, just exiting on opposite sides). If the channel is large enough, you can use a small safety pin attached to the end of the cord & slide it through that way, that's easiest.
On the ones I've made, I make them lined, with the last opening on the bottom inside part. That gets closed too, but all the other seams are hidden and out of sight this way. The cord gets sewn in on the side of the bag and I tie them closed rather than doing it the drawstring way where you just pull. It cuts down on the sewing/putzing time for each bag by quite a bit when you do it that way. I've heard of pouches made with a top flap that goes down and attaches with a button, snap or velcro, too.
Check the fabric at the store. If it's sheer, it'll probably catch on something & tear at some point, those just don't last long. If it's able to be snagged, make sure its not the kind of fabric that will 'run' like a snagged nylon stocking, I think that problem is mostly with knits. And if it's an unusual kind of 'homemade' fabric like some of the batiks that have hit the stores lately, check to see that the dyes are permanent, I had a blue and a red batik actually rub off colors on my purse and jeans, which was very unexpected. I think those particular batiks came from India, it was a special purchase at the store. I didn't pre-wash it for the project I'd used it for, so it was probably my fault.
Oh, and check the cording you want to use too, some start to fray on the ends when cut, you'll have to tie a knot in the end to stop it from destroying the last 3" of the cord ends. Also give them a good tug in the store before buying, some don't stand up to pulling too well and can snap.