Build a 5 mile tall building (or taller) on the equator. The building is basically a giant rail gun. In other words, use the same technology the military ships are using to launch projectiles 17 miles away, except this gun shoots a rocket towards space.
This building/rocket launcher could have two settings; one would be with hyper g-force to shoot inanimate stuff into space like water, extra fuel, structural parts, and dirt for a massive space station or lunar colony. I imagine this setting could probably get something into a low earth orbit without using any propellant. The other setting would setting would be a slower, non-lethal, and lower g-force setting for shooting people into space.
The building shoots objects many miles towards space and rockets kick-in at some point and continue to propel the object the rest of they way into space.
The reason I say build it on the equator is because that's the area of the earth that spinning the fastest already.