This is indeed a bit confusing. Supply and demand functions depend on potential prices. That is, the agents think "if the price were a given level, what would I want to supply or demand?". So supply and demand depend on a whole range of (hypothetical) prices. However, the equilibrium price (which is a certain price and different from the range of potential prices) is that price where both supply and demand meet. In your example, the equilibrium is where the circular logic stops and finds a resting point, which is also why we refer to it as an equilibrium. This is a concept, which we borrowed from physics and is often used to mean a position where things stop changing/moving.
You are right to ask, however, how this whole process is initialized.
The rigorous way to think about this for economists, is to use what is called a "Walrasian Auctioneer".
You can imagine the process as such. An auctioneer calls out prices. Suppliers report how much they wish to supply, whereas consumers report how much they wish to demand. If these do not meet, the auctioneer calls out other prices. This goes on until demand meets supply and that's when the auctioneer stops.
Of course in reality, there is no walrasian auctioneer. Nevertheless, it is not hard to imagine that a new firm sets some price when it starts and then adjusts it depending on demand. Furthermore, depending on that new suppliers might enter, which changes supply and causes the price to adjust again. This goes on until a stable point is found, which we call the equilibrium.