The only way to get from Earth to somewhere else is with a big, highly-optimised rocket, because anything less cannot achieve escape velocity.
Obviously, any civilisation capable of sending enough off of Earth to succeed in spreading off planet must have large scale industry and pretty good technology.

If Earth was even bigger, rockets as we know them wouldn't be enough, because the energy density a rocket would need for escape velocity would be more than that of fuel so even if a mass of fuel somehow flew it couldn't leave Earth.
In that case, what's needed is either supermassive scale industry (like a stupidly huge railgun), or very advanced technology.

If Earth was smaller, the amount of fuel spacecraft would need for escape velocity would be drastically lower; not only does halving escape velocity quarter energy requirement, needing less energy means you need less fuel which means you need less energy so making Earth significantly smaller could mean that just sticking auxiliary rockets on a plane would be enough to make it a spacecraft.
In that case, all that's needed to leave Earth is WW2-era aircraft and rocket technology; a spitfire pilot with enough supplies and maths skills could just leave Earth and do a flyby of the moon, assuming their spitfire was equipped with rockets (sticking rockets on an interceptor so it can reach battles like an ICBM would be commonplace on a smaller Earth).

If Earth was smaller, and a moon around a gas giant, not only would leaving the planet be trivial; the logistics of establishing colonies would be trivial as well since Earth would be a stone's throw from many viable colony sites.