Welcome!
I started out in my kitchen and then my Hubby built me a small out-building ( soap factory!) on our property. That was nice and clean and we never let in animals and had good screens and screen doors. That was then.
Now, I am starting over again in my sister's garage - not so clean in there.
I just make sure tools and everything are washed and clean and then after I pour my molds I cover them right away. I also cover all of my stuff in between uses with plastic sheets ( cheap painter's drop clothes.) but I clean everything again before and after use. I haven't had any strangeness in any of my soaps.
If you want to put up walls and a ceiling and cover them in linolium or tile? That would be a true clean room - but you're not selling food, are you? CP soap doesn't need to be sterile - the lye kills a lot of stuff!
Just my 2 cents. Hope that helps.