NASA’s DART mission flew on a falcon 9, but it flew 7 million miles. That’s a lot further away than the moon is, so why would falcon heavy or starship ever be necessary to go there? Why can’t people just go to the moon in a falcon 9? I’m sure there’s a very good reason, I just don’t know it.
For a given destination - the moon, Mars, or a particular asteroid - you need a rocket sized proportionally to the payload you want to send there. DART was a very small payload -- only 610kg of mass. To land on the moon and return crew safely, you need something like 50 to 100 times larger payload (depending on how many people you want to take and how long you want to stay), which means you need a much more powerful rocket.
First: Falcon 9 is just a launch vehicle. DART is a complete spacecraft in its own right, with its own guidance and propulsion systems. (It was actually two spacecraft: DART itself was the impactor, and it deployed the LICIACube cubesat to observe the impact.) If you just duct-taped a human to the top of the Falcon 9 second stage, you could get them pretty far, but they'd die not long after liftoff.
SpaceX operates the Crew Dragon to put people into low orbit. Where DART+LICIACube together massed 624 kg, a Crew Dragon with 4 crew and 150 kg of supplies masses 9616 kg. That's over 15 times as much mass. Falcon 9 just can't throw that much mass to the moon, and the Dragon wouldn't be able to land on the moon when it got there.
You need a spacecraft capable of maneuvering, landing, and returning the crew. That will all take a much larger spacecraft or a combination of multiple smaller ones, and a much larger rocket to get it on its way to the moon.