You are legally recognized as a business the moment you earn a dollar doing business. You're called a sole proprietor, which is a business doing business under the name of its owner. Taxes and legal obligations pass through directly to you.
There are other types of business entities you can form -- partnerships, LLCs, S-corps, C-corps, or a simple DBA ("doing business as") -- which can have both benefits and downsides. You can talk to either a CPA or a lawyer that specializes in small business about what makes sense for you.
The next and easiest 'upgrade' from sole proprietor for most people is likely an LLC. As a sole proprietor, your personal assets are at risk in the event of a lawsuit or class action and such. Incorporating in some fashion will protect your personal assets in such an event and is arguably a more direct upgrade from sole proprietor tax-wise.
It's only your call when to 'upgrade' if ever, but the answer is generally somewhere between 'getting traction' and 'getting sued', with the real kicker that you obviously can't incorporate after getting sued so, y'know, there's that.