Just for the basics that you need to get started. A lot of people start with the mixing bowls from the dollar store, and they work well enough. I found the handle to be a little flimsy, though and bought
this bowl which I like. The measurements on the inside have stood up to the raw soap batter and I've been using it for 4 years now. It looks like the one I have has been updated (mine had red printing on the inside of the bowl), but it is an OXO brand. For splitting off batter to color, a plastic 4 cup measuring cup (I bought mine at Walmart for under a dollar) works well.
I prefer one piece silicone spatulas, and love the spoonulas and spatulas from Micas and More
Supplies Archives - Micas and More Amazon also has a lot of similar choices.
My stick blender is a very old Braun that hasn't been made in years. On the recommendation of a few members here, I ordered
this one and I love it. I didn't think I'd find one that I like as well as my old faithful, but I do! You want the 2018 model without the button on the top if you decide to get one. There are more economical choices, and a search here will give you lots of threads to read.
I use a dedicated stainless frothing pitcher for mixing my lye solution, but when it is dirty I have found that the Ziplock Twist and Lock containers work very well. If I'd known, I might not have bought the stainless pitcher. They are available at Target and grocery stores.
For a mold, you can use anything - a sturdy box lined with freezer paper works well. There are some inexpensive silicone options on Amazon, or soap supply vendors sell them. A small one to start with is wise so you can make smaller batches. 1-2 lbs (of oils) is a good size.
This mold works well, and you can use it for test batches going forward. It's available at Bramble Berry and Wholesale Supplies Plus as well (probably more places too), so shop your best price or add to an order if you are placing one somewhere else.
That will get you started, other than your PPE - gloves and goggles at the least. Ask if you are wondering about anything else or if something is unclear. Lots of people here with great advice.