In order to do hatha yoga you need not buy special gear. You can do yoga on the grass, sand, or a broad flat stone under trees or under the sky. You can do yoga on the bare floor, a rug, a carpet or seated on a chair or lying in bed.
But sometimes its nice to have the trappings. A sticky mat makes triangle poses feel safer. Sometimes it's nice to have a block or strap to make some poses easier. This page offers suggestions and a portal to exploration of the options that are out there.
If you need a mat, consider getting one that is ecologically friendly. They don't have to be spendy. You can compare those with mats that make no environmental claims but are pretty and practical. Some mats you can wash in your washing machine. Others need special care. That's why the mat cleaner is included above. You can also use a weak vinegar solution for mat cleaning.
Choose your block according to your use. They vary by material and thickness so some are softer, some firmer. Since some people like to sit on their blocks, thickness can be important.
Straps can be a big help if you have tight hamstrings or cranky knees. A nice long one is great for bound angle pose. Keep in mind your own size. Most people can manage with a six or eight foot belt for that pose. And those are the most common lengths of yoga strap. A few of us need ten feet.
Any old belt you may have will work for most things. Choose soft leather or woven. I was lucky enough to get a batch of old Nicaraguan belts that are lovely for most purposes.
Some of us have weak wrists or wrists that don't bend quite enough for poses like the downward facing dog. If that's you, try pleating the end of your mat to make a thicker cushion and let your fingers fall off the front of it. If that doesn't work or is too fussy for you, see if some of these options will work better.
You probably don't want to tote a bolster with you to class, but for home practice it might be nice. There are a number of stress relieving poses that a bolster would be a big help for. You can, of course, use a rolled up blanket instead.
Blanket: Yes, you can cover up with it or roll it up to support your knees during meditation. But the thing blankets are really great for is during the shoulder stand. If you feel a stretch at the back of your neck, that's a bad thing. You could damage your neck. Fold the blanket up and position it so that when you roll up onto your shoulders your head and neck hang off the front edge of the inch or two inch thick folded blanket.
The best blanket is firm, and not squishy soft. Flat woven cotton is good. So is wool. You may already have a good blanket at home. If not, check out Amazon.
Equipment Bags: I use a roomy duffle bag to tote my yoga traps to and from class. You may prefer something designed for the purpose.
Eye Pillows: Sometimes the room is just too bright for comfortable meditation. Of course you can drape a towel or the sleeve of a sweater across your eyes. But you can also bring an eye pillow!
Yoga Socks and Toe Stretchers: Yoga socks have a little sock cell for each toe. They often have non-skid coating on the soles so you can get along without a mat. Since toes are so often crammed into shoes and healthy toes need to stretch out now and then.
Towels: Beyond wiping your dewy brow, you can place one on top of your yoga mat, or use it when you don't have mat. Some of these towels are made with microfiber to keep you from slipping and are super-absorbent.