If you are asking how to become a legal entity, you incorporate through your state's Secretary of State. For example, my Rusty Hills Farm is legally incorporated in the state of Alabama. We pay both state and federal corporate taxes and file all the proper tax and reporting forms. We have a corporate bank account to keep the corporate funds completely separate from our personal funds. We keep a legal set of books for the corporation and maintain a file of all our business receipts and income records. We show a profit at least 3 years out of 4. However, incorporation is not always your best financial option tax-wise, so you should really talk to an accountant before you take the plunge
Any corporation can pretty much be formed under any name you want--as long as the name is available (ie nobody else has already incorporated using that name) and your Secretary of State approves it. I'm not aware of any limitations on calling yourself a farm or a ranch--think of all the housing developments called "Happiness Farms" or "Rocky Mountain Ranches" or something similar and they certainly are not engaged in agriculture!