The most expensive piece of equipment will be the scale. You need to have a digital or electronic scale in order to get accurate measurements. My first batch of soap was way too high in PH when I used a spring loaded type of kitchen food scale.
The links below will give lots of good information in detail about the basic needed supplies. In addition to those, you will need a few kitchen-type utensils. Whatever supplies you use, have them designated for soap only, and not return them to food or kitchen use. A good place to get these really cheap is a dollar store or thrift store.
The following cold process soap making equipment items have pages of their own:
Make sure to check out the other pages regarding soap basics, which could include coconut oil. Olive oil only soap makes for a small creamy lather, whereas the addition of 10 percent coconut oil makes a bubbly bar.
From "cold process soap making equipment" to a super moisturizing addition - cocoa butter...
If you are just starting making soap and need a company name, check these out...