Assuming the civilization is willing to spend the resources necessary to evacuate Earth, the best way to do this using current technology is to build a laser or microwave launcher.
This system (first proposed by Arthur Kantrowitz and Wolfgang Moekel in the 1970's) uses the energy of an off board generator to energize the reaction mass and can provide extremely efficient rocket propulsion (Lyek Myrabo's "Lightship" proposals using atmospheric gasses for the first part of the flight have an effective ISP of infinity, once past the sensible atmosphere and using laser light to heat H2 reaction mass the ISP drops to a "mere" 1000 seconds). Since the rocket does not have to carry fuel and oxidizer, and the rocket motor can consist essentially of a parabolic reflector, the rocket is much smaller and lighter than what we are used to (the Space Shuttle launch stack of 2000 tons was mostly propellant and liquid oxygen; the shuttle itself weighed @ 120 tons fully loaded).
Laser launchers could potentially operate on an assembly line basis for evacuation, being build, loaded and rolled straight to the launch site and launched with minimum preparation. Since the preparation and handling of spacecraft is the primary driver of costs, (fuel and liquid oxygen are pennies to a few dollars a kilogram, and the hardware is about as expensive as a jumbo jet per kilogram as a rough estimate) minimizing this makes launching much cheaper.
Microwave beams for rocket launching work on similar principles. The microwave generator is potentially much cheaper and more efficient than lasers of equal power, but the means of coupling the microwave energy to the reaction mass of the spacecraft is different and the spacecraft itself might be much larger to carry the microwave receiver; the tradeoff in this case might not be worth while.
Once everyone is in orbit, the laser beam from Earth could be redirected by a series of orbiting mirrors to power the spacecraft around in the Earth Moon system, but if we are going that far, the first wave of people into orbit should have been building proper infrastructure to live and move in deep space.