You don't have to be a "production company" to make films. You
should have a business entity separate from your personal identity to protect yourself legally.
Start with your local Small Business Association. They will get you started setting up a business. It can be as simple as a “DBA” (Doing Business As) or an “LLC” (Limited Liability Corporation). And then there are many other variations. Learn about them and choose the one that suits you best.
A DBA - which in my area means a commercial bank account and a $10 fee at the town hall - establishes that you are a business but provides nothing in the way of legal protection. An LLC provides liability protections - for a price - and requires you to do a fair amount of paperwork. A common practice is to have an umbrella corporation that provides the framework for all of your business activities. You then create separate LLCs for each of your productions. Sometimes you also create separate corporate entities for each aspect of your company's operations.
For instance, LucasFilm has Skywalker Sound, Industrial Light & Magic, Lucas Licensing, Lucas Learning, Lucas Books, LucasArts and Lucasfilm Animation under its corporate umbrella. Each subdivision takes care of its own management, taxes, employees, etc. but are responsible to - and overseen by - the parent corporation. To top it all off, each film produced by LucasFilm is a separate LLC created exclusively for the production of that one project. In essence, "Indiana Jones 5, Geriatrics Revenge LLC" is legally a separate corporate identity that "hires" LucasFilm and it's subsidiary units to produce the film. In this way the only corporate entity liable for any litigation against the project is "Indiana Jones 5, Geriatrics Revenge LLC"; none of the other corporate entities have legal liabilities connected with the production.
It's a lot more complicated than that, but we'll leave that to the lawyers; it's their job to strain at gnats, swallow elephants and to shove the occasional camel through the eye of a needle. That being said, you should
always consult an attorney when making legal changes to your life.