Updated: May 19
It takes a while learn exactly what you want in your yoga kit bag.
There are certain items that I'd recommend as essential for your comfort and safety such as a non-slip mat, a blanket and some water.
We often use yoga straps for certain exercises so for hygiene it's best that you have your own. In my classes we use both the stretchy resistance belts and classic cotton yoga belts.
My personal yoga kit has a yoga mat, two full sized yoga blocks, two half sized yoga blocks, a small bolster, a yoga belt, a resistance belt, an eye pillow and a blanket.
The blocks are a good idea as the provide support in so many poses. Things like bolsters and eye pillows are individual comfort choices. If you have low back pain, then a bolster or a bed pillow are great props to get you comfy in relaxation.
Thanks to COVID 19, we no longer share props, however as you probably wouldn't go swimming without your trunks and people who have kit are more likely to practice at home, it's a good thing to have your own kit. And more hygienic anyway!
A final note about mats. Ideally you want something not too thick (hard to balance in standing!) and not too thin (uncomfortable!). The mat I use in class is 183cm x 61cm, 5.5mm thick which I find 'just right'.