First, the fun part.
The favorite simulator of amateur rocket scientists is Orbiter . It simulates the experience of flying a space ship, with realistic rocket science behind it, but with none of the expense or danger. Orbiter is the creation of Dr. Martin Schweiger, a professional scientist (but we don’t hold his lack of amateur standing against him). He has invested untold hours of effort since 2000, and released the program for anyone to use for free. The program has been improved several times over the years.
Dr. Schweiger developed an 'engine' that can simulate the gravitational interactions between the Sun, the planets and their moons, and a variety of spacecraft, as well as the interaction of spacecraft with the atmosphere of Earth or other planets. He also developed a visualization system that displays the planets and spacecraft as if you were in the commander’s seat, or drifting outside. Perhaps just as important: he included the 'hooks' that allow Orbiter enthusiasts (called 'orbinauts') to add new spacecraft, instruments, spaceports, visualization clients, planets or even whole solar systems to extend the fun.
You will find the instructions for obtaining and installing Orbiter, and for running a demonstration spaceflight, in Installing Orbiter. The traditional introduction to enjoying Orbiter is Bruce Irving's Go Play In Space; a quick look might convince you that Orbiter is worth investing a bit of your time.
The more experienced members of the Orbiter community are welcome to further develop an approach to Rocket Science for Amateurs, and add to this description.