I've been cultivating recreational, medicinal and gourmet mushrooms for years. The equipment needed for tissue culture is the same as mycology. Keeping things low key and basic you'll need a pressure cooker and a still air glove box, eventually upgrade to a laminar flowhood vertical is what i have, but a horizontal will work the same. I love my induction sterilizer but an alcohol lamp will work. Stainless steel tools are best but you can get away with disposable. Glass petri plates can be reused and sterilized, plastic are junk and most times not sterilized regardless of what they say. I'm not sure about the matrix you'll be needing but I'm pretty sure agar is what you will be using, depending on the development phase your targeting, hormones are needed to be added to the agar. And finally... glass jars to accommodate for vertical growth when they outgrow the petri plates.
This is an area I've been interested in for years, just never got around to it. I've got my hands full with everything I'm doing now. Good luck, your about to be sucked down a rabbit hole from which you'll never return. I absolutely love mycology, it's so rewarding, and I may have went overboard building a clean room and the hepa filtration, but cleanliness is key and sterility is king. You'll come to find all your "weakest links" that add to failures, 99% of the time.... it's sterility and cleanliness.