How fast do astronauts travel to the moon?

There is no cruising speed on the way to the Moon, because nothing is thrusting. A lunar spacecraft accelerates hard for a few minutes, then coasts for the rest of the journey, gaining and losing speed exactly as gravity dictates. Apollo 11 left the pad on 16 July 1969 and landed on 20 July, so the 384,400 km that separate Earth and Moon on average were crossed in a little over four days from lift-off to touchdown, a mean pace below 4,000 km/h. That figure describes almost no moment of the actual trip.

The profile is the interesting part. The burn out of Earth orbit, the translunar injection, is where the spacecraft is fastest; from then on it is climbing out of Earth's gravity well and losing speed the whole way, like a ball thrown upwards. Because Earth outweighs the Moon by a factor of about 81, the point where the two pulls balance lies roughly nine tenths of the way across, and by then the vehicle is crawling. Then it accelerates again, falling towards the Moon, and must brake into orbit or sail past.

The return is the mirror image, and it ends with the only genuinely high speed of the trip: a capsule coming back from the Moon meets the atmosphere far faster than one returning from low Earth orbit, which is why lunar heat shields are a category of their own. Artemis crews fly the same arithmetic, with outbound legs counted in days rather than hours, and nothing on the drawing board changes it. A shorter trip is possible, but every hour saved is bought with propellant on departure and paid for again on arrival.