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You can't point a rocket at Mars and fire. Mars won't be there when you arrive. Spacecraft ride a carefully timed half-ellipse around the Sun, and the timing is set by the planets, not by you.
It's like a quarterback leading a receiver: you throw to where Mars will be. The trip takes a fixed 259 days, so Mars has to start about 44° ahead of Earth. That lineup only happens once every 26 months, and that's what sets the launch date.
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Model: circular, coplanar planetary orbits and short (impulsive) burns. That's the textbook Hohmann transfer. Real missions use the planets' actual elliptical, slightly tilted orbits, so real launch windows can shift by a few weeks and real delta-v budgets run a bit higher. Getting off the ground and into the parking orbit (about 9.4 km/s from Earth's surface) is not included. Pluto's orbit is far more stretched and tilted (17°) than the planets', so treat its numbers as rough. The Moon option is worked out around Earth, starting from low Earth orbit.
In 1925, German engineer Walter Hohmann worked out the cheapest way to move between two circular orbits: fire your engine once to stretch your orbit into an ellipse that just touches the destination orbit, coast halfway around, then fire again to round the orbit out. The transfer ellipse is tangent to both orbits, so neither burn wastes any effort pushing sideways.
The catch is that the travel time is fixed by the orbit's size, not by how hard you push. Earth to Mars on a Hohmann transfer always takes about 259 days. You can get there faster by spending more fuel on a steeper path, but you can't get there cheaper.
Because the coast takes months, you have to aim where the destination will be when you arrive, not where it is when you leave. For a Hohmann transfer to Mars, the spacecraft sweeps 180° around the Sun. In those 259 days, Mars moves about 136° along its own orbit. So at launch, Mars needs to be about 44° ahead of Earth. Launch any other day, and you get to Mars's orbit while Mars is somewhere else.
Earth and Mars only line up that way once every 26 months. Miss the window and you wait over two years for the next one, which is why every Mars mission of a given era (rovers, orbiters, landers from several countries) tends to launch within the same few weeks.
The same logic applies coming home. After a 259-day trip out, the planets are in the wrong places for the return trip, so a crew has to stay at Mars for about 15 months waiting for the return window. Round trip: roughly 2.7 years. The calculator above works out the wait for any pair of planets.
North, south, east, and west only mean something because you're standing on a spinning ball with a magnetic field. In orbit there's no ground to point at. Pilots and mission planners use directions defined by your own motion instead, relative to whatever you're orbiting (the Sun, for an interplanetary trip).
Those six directions are all a spacecraft ever needs, and they lead to some of the least intuitive rules in physics:
Even "up" is just a convention. For the solar system, astronomers call the plane of Earth's orbit the ecliptic, and by agreement "north" of it is the side from which the planets appear to circle counterclockwise. Nothing out there marks it; it's a choice of which way to hold the map.