Some places have props, some don't. Some classes use props, some don't. (If you are in a place without props, the class will tend not to use props!) I'd wait a bit on a bag and see what you need. Otherwise you can easily wind up with an assortment of special purpose bags you don't like or use. Don't put anything in plastic.
Since I rarely use blocks I don't worry about taking any along to places that don't have them and rarely does anyone else I see, but if I wanted to I could easily toss them in a random tote bag. Mine are just decorating a corner at home, they go nicely with the carpet.
For my mat I prefer strap type holders to bags, my favorite mats are heavy and sticky and tend not to go in the bags well. I made a mat carrier out of a yoga strap and a couple quick release buckles, works great and I can still use it as a strap. The buckles actually make it easier to hold onto. I used to use a strap carrier that closed with velcro, and I hated how loud it was when I opened it in a quiet room!