To be fair, falcon 9 also gets refurbished between flights.

But it's way less time/labor intensive than the shuttle.

The external tank and both SRBs could be refurbished and reused. The orbiter also forms part of the first stage.

> From my elementary school memories, the Space Shuttle used a giant center rocket and two booster rockets that we expendable

That's incorrect. The orange "giant center rocket" had no engines, it was an external tank which held the propellant burned by the orbiter's main engines. The SSMEs were on the "Shuttle thing" and were reused.

The space shuttle had three main components - the orbiter (white and black space plane with the crew and cargo), the external tank (rust brown on all but the first few flights), and a pair of solid rocket boosters. The orbiter had three engines and received its propellant (oxygen and hydrogen) from the external tank. At launch, the solid rocket boosters (SRB) and the main engines were all running so technically they can all be considered part of a “first stage”. The SRBs would expend their propellant in around two minutes and separate. They dropped into the ocean, slowed via parachute, and were recovered by boat and refurbished. The three space shuttle main engines would continue to burn for several more minutes until the external tank was depleted. The ET would be discarded and burn up on reentry. Then the orbiter would use two small rockets with internally stored hypergolic propellants to boost up to its intended orbit.

Apollo/Saturn had a much more traditional staging design where the first stage booster would run then drop off, then the second stage booster would run, then drop off, etc. There exist other rockets like the Atlas which had what they called a “stage and a half” design where the center stage burned for a long time and there was an outer “ring” stage that dropped off after a shorter time while the center stage kept going.

After going to the moon, the only part of the entire Apollo/Saturn rocket that came back was the command module capsule with the astronauts and moon rocks. These were completely torched and affected by salt water and were not designed to be reused. Reentry from the moon is significantly faster than reentry from low earth orbit.

The shuttle orbiters were refurbished and reused. It was a very difficult, expensive, and time consuming process to get an orbiter ready for another flight.

All of that is to say that the picture is blurry for the shuttle regarding first stage reusability. Yes the SRBs and orbiters were reused and were lit at launch. The ET was in use at launch but discarded. In my opinion it’s sort of not that interesting to argue what is and isn’t first stage on the shuttle because the elements just don’t map cleanly.

The design of the system with the SRBs, tank, and orbiters being adjacent to each other is considered by many to have been too dangerous in retrospect. This design was a factor in both shuttle disasters - the SRB shooting fire at the ET causing it to explode on Challenger and cracked foam falling away from the ET during launch and hitting the leading edge of the orbiter wing for Columbia. If they were stacked vertically rather than adjacent, those specific failure modes would not have been possible.