Do I need to buy specific props to do these classes? NO.
Read that again - Do NOT buy anything until you try out the exercises with what you already have in your home. Everyone invests in props at different times based on their finances and interests. Below I share what I would want to know from my coach about purchasing a prop. Full disclosure, I get commissions for purchases made through these links. When someone asks me about a specific prop, I always recommend first using what you already own and have around your home. Look for the Don't have any? suggestions below. Next, when you are ready to buy, you will be better informed by your own experience to evaluate my suggestions.
The absolute best not slip yoga mats I found are by Jade brand.
Yoga Outlet typically has sales when you are ready to invest. Paying $60-$100 is reasonable for a high quality mat, but you definitely do not need to spend that much. Less expensive options? Great mats can be found at Target and local Dollar Stores. Look for a thin and sticky type mat so that it doesn't slip. The gymnastic or pilates thicker mats tend to slide too much and when you are trying to balance or put sweaty hands on it, they simply don't work. Don't have any? Try a beach or bath towel or directly on non-skid flooring in your home.
The 4" foam blocks by Gaiam are pretty popular. I find the 3" blocks are too wobbly when used for balancing poses. Cork blocks are more like light weights because they are heavy and hard to grip, and because of that, I would NOT recommend cork blocks for beginners. Foam blocks are preferred. Again, Yoga Outlet has a variety of colors and style when you are ready to invest. I recommend everyone get two (2) blocks to use. Don't have any? A step stool, a thick hard cover book, bath towel, or even a chair and wall work well in place of blocks depending on the exercise.
A strap is great for times when your hands don't quite reach your feet or you want to add extra focus on your spine and not worry about how far away you are. Often local stores carry yoga straps for reasonable prices. Any material will work, just make sure to get one long enough for you. You can always shorten where you grip the strap, but if you are tall, have long legs and short arms, or lack mobility, then I'd suggest a 6-8' strap. Yoga Outlet has many sizes and color options. Don't have any? A bathrobe strap or waist belt will do in a pinch!
I suggest getting a light resistance band that is about 6' and can be stretched long (not a circle) like this one from Yoga Outlet. These can also be found at Target and other local stores. I use resistance bands to provide extra work to regular yoga poses and light stretches when I'm ready to up my level. I also use them to up my level of awareness by holding a band around my rib cage and noticing where my breath goes. Don't have any? You can definitely use a strap or the substitutes mentioned above for straps. They just won't be stretchy and you will need to move your hands/body a bit differently.
One of the main props for Restorative Exercise classes is the half cylinder. You can purchase this from my mentor, Nutritious Movement for about $10 + shipping, or if you are local to Rochester purchase directly from me for $5 (via PayPal or cash) and avoid shipping costs. There are so many ways I use the half cylinder in classes, if you make one investment, this is it! Don't have any? Take a bath towel and roll it up tight.
YOGA TUNE UP BALLS
You can use soft, round balls for self-massage techniques. Use them under feet or glutes or shoulders. Tennis balls are the least expensive and you can usually get these from Dollar Stores. For the rubber balls which grip onto your skin more firmly rolling the tissues, then try Yoga Tune Up Therapy Ball Pair in Tote (about $12) or the Yoga Tune Up All 3 Sizes (about $48). Don't have any? Use a pair of thick socks rolled up into a ball.
BOLSTER - FIRM PILLOW
Using a firm bolster for comfort and support can make a difference in your ability to both relax and engage muscles. This example from Yoga Outlet is what I would look for - something rectangular in shape and a few inches high. Local yoga studios may sell some as well, but I would be leery of bolsters at Dollar Stores simply because their filling will deteriorate rapidly and not offer you much 'bolstering' with use. Don't have any? Use several folded blankets or bath towels.