Seeking deeper understand of reading a schematic with an op-amp IC (AD-8574)
Is that IC in the pinout a quad channel operational amplifier?


Figure 1. It's written so big that you missed it!
Yes. It's a quad amplifier.

... is the schematic indicating the actual pin numbers to run the traces to?

Yes.

Seems like a clearly defined blueprint of what exactly I should be doing or am I just being overly optimistic?

The schematic shows the schema and good one's make it clear what each section is supposed to do. At high frequency layout becomes important as stray capacitance and inductance can affect the signals. This circuit will probably run fine on breadboard or prototyping board if you're running, say, at audio frequencies or less.