Icm -
I'm really curious about how you are turning t-shirts into swim suits.
Generally, the reason nylon, lycra, etc are used for swimsuits is because they dry quickly. However as long as the shirts are lined, cotton should work just fine. I'd recommend suggesting that people rinse their suits after swimming with fresh water (which is a good idea no matter what the suits are made of), because I think cotton might be weakened more by the chlorine in pools than some other materials. (On the other hand, Stretch-and-sew recommends using cotton covered elastic for swimwear, so maybe not.)
As for sewing swimsuit elastic on, it's pretty much like any other covered elastic. You sew a piece together to form a loop, pin it at equal intervals to the fabric you are sewing it to, stretch the elastic to the length of the fabric and sew it on. Then you fold it over (once or twice depending on your pattern/goal), stretch it out so it is smooth, and sew it again.
You can often find outdated styles of Stretch and Sew swimsuit patterns in thrift stores very inexpensively. They have good directions useful for general swimwear construction.
Julie