Plants are designed on paper (and in software) long before steel is cut, through a series of drawings that each add detail. A block flow diagram shows the main steps as boxes: feed preparation, reaction, separation, product. It's for the first big decisions.
A process flow diagram (PFD) shows every major piece of equipment (reactors, columns, exchangers, pumps, tanks) with standard symbols, numbered streams joining them, and a stream table listing each stream's flow, composition, temperature and pressure. That table comes straight from the mass and energy balances.
A piping and instrumentation diagram (P&ID) is the detailed drawing the plant is built and run from. It shows every pipe with its size, every valve, every instrument and control loop, relief valves, drains and vents. Instruments are drawn as circles with letter codes: the first letter is what's measured (T temperature, P pressure, F flow, L level), and the next ones what the device does (I indicate, C control, T transmit). So TIC is a temperature indicating controller.
Design usually works from the inside out, like the layers of an onion: choose the reactor first, then the separation and recycle system, then the heat exchanger network that recovers heat between streams, and finally utilities (steam, cooling water, power). Each layer depends on the ones inside it.
Today most of the balances are done in process simulators, but engineers still check them by hand. A quick overall balance, like the ones in the first chemical engineering course, catches most mistakes in a simulation.