Why Project Gemini was Critical for NASA's Moon Landing

Project Gemini was designed to develop and prove the technologies needed in order to go to the Moon.

From: SPACE VOYAGES: Into the Unknown

[MarsFKA]

The Gemini 6 launch was delayed three days because of a fault during the first launch attempt. Actually, calling it a "fault" is about the same level of understatement as calling World War 2 a "heated argument".
Gemini 6 reached zero in the countdown and the engines fired. Then they turned off and Launch Control had a group cardiac arrest because their instruments showed the the rocket had moved, that is, begun to lift off before the engines shut down. The only thing that would happen after that was a huge explosion, with, hopefully, the crew escaping in their ejector seats.
Meanwhile, in the space craft, the astronauts had felt the engines start, then shut down, but they had not felt the rocket move. Schirra, the commander, was sitting there holding the abort handle and his brain going If he pulled the handle and they ejected, that was the mission done and dusted, right there. As well, it would probably mean their astronaut careers were done and dusted too, because those ejector seats had tremendous acceleration and any ejection was likely to cause some kind of spinal injuries - many aircrew have had their flying careers ended by ejections that otherwise saved their lives.
In the event, nothing happened. The rocket sat where it was and Schirra, a veteran test pilot, was acclaimed for keeping his cool under unprecedented conditions, although he later admitted that those few seconds, while he sat on a 153-ton bomb trying to decide whether or not to blow himself and Stafford off the top of it, were the most frightening of his life.
The problem was traced to a cap that had not been removed from a plug, fooling the rocket's computer into thinking that all the fuel had been used up and that it was time to shut the engines down.

[SmithsonianChannel]

Major Edward White reminded us that no dream is too big, when he became the first American to walk in space on this day in 1965.

[josephastier]

We should have continued the Agena vehicle development until it was most of the Saturn V. Once in orbit and checked out, the crew could fly up in the command module, dock, and do another systems check. Then hit the gas for the Moon. Doing it as two vehicles would have meant a greatly simplified development process. Most of the Moon missions would have been cargo flights, with only a very small part of it man-rated to LEO.