I do wonder how easy it was for the astronauts to take off from the moon, in their landing craft. I realise that there's no atmosphere there, and that gravity is about one sixth of that of the Earth's - but still, from what little I know, this particular part of a "moon journey" never seems to be particularly troublesome. Can anyone explain why this is so?
The amount of delta-V needed to get from the Moon to low Moon orbit is only 1.87 km/s or about 1/5 of that needed to get from Earth to LEO. That amount is easily attainable even if your ascent vehicle has a high payload fraction (i.e. only a small part of your vehicle is fuel). Other sections of an Earth-to-Moon-and-back mission require much more delta-V.
This also meant the ascent engine could be designed for simplicity and reliability, instead of ultimate performance. It used hypergolic propellants, so no ignition system was needed. It was pressure-fed, so there were no turbopumps. All you needed to do was open two valves, and the engine would run.
What also helped was the Lunar orbit rendezvous technique: the Moon ascent stage has no other job than to get two astronauts and their cargo to Moon orbit, so it can be small and light.
In the original mission concept, the Earth reentry module would also be used for the Moon landing. This meant landing a much heavier module on the Moon. It also meant this heavy module had to take off again not just to low Moon orbit, but on a Moon escape trajectory to Earth. This concept would have been too heavy for the Saturn V and needed a much larger rocket.