I want to sew together a flag and I was wondering what type of fabric would be best to use. I'm looking for something opaque, thin, non-stretchy, and available in various colors. I initially thought cotton or fleece would be good, but cotton is too stretchy and fleece is too thick.
Most commercial flags are made out of nylon. It's lightweight, durable, and dyes well. Specifically, the flags on many sites are made from DuPont SolarMax™, which is supposed to be nicely color safe in the sun. For inside use, you should probably be fine with a ripstop style nylon.
This heavyweight DuPont SolarMax™ nylon is the most popular and versatile flag fabric available. Nylon's combination of strength and brilliant display, along with its quick drying ability, make it suitable for a wide range of applications. Great in rainy areas, its light weight and close weave enable it to fly in the slightest of breeze, giving the fullest visual effect. Nearly every flag we sell is available in nylon material.
Other options include various types of polyester, which is unlikely to stretch the way the cotton you've been using has,rayon (which is a cellulosic fiber like cotton, but does not have the strength of cotton), and the right type of cotton, which is apparently a "2x2-ply mercerized cotton", which is specially prepared in such a way that it absorbs more dyes, is more lustrous, and stronger, so it is less likely to stretch or deform.
I suggest looking at the lining fabrics in a fabric store. These fabrics are usually in a section all together, and grouped by color. They are most often 100% polyester which is very strong, and for this purpose, light-weight (lining fabrics are an extra layer of fabric inside a clothing item, like a dress or a suit, to add strength or protect from stretching or to prevent see-through). Lining fabrics come in a broad range of colors from pastels to vivid brights. They are woven fabrics (vs. knits) so they don't stretch. They are usually very colorfast, and dry quickly if they get wet. They may be thinner than what you are looking for, but they will flap nicely in a breeze.