Sounds like a system of space planes that operate in the style of the railway system in the early 20th century. Since you said you already had space elevators in place, then you would need a series of space stations that act as rail stations near the elevators. The plane would dock at the station, which gives you quick turnaround time since they never enter an atmosphere. Since everything is in space the space station could be multilevel allowing several planes to dock at once.
As far as propulsion, I don't have an exact method, yet, but I know how to meet the "2 days to Mars at closest approach" requirement. An engine that is capable of sustaining 1g (Earth normal gravity) of constant acceleration will get a plane from Earth to Mars, assumed 65 million km separation, in 1d 21h 13m 1s. I would love to take credit for figuring that out, but alas, I can't. Check out How fast will 1g get you there? for a really great set of charts, graphs, and travel times. Somebody figured out all the travels times from Earth to each major body in the solar system. 947
That time is for constant acceleration for half the trip, then the ship flips a 180 and does a constant burn for deceleration.
Another benefit of constant acceleration is that it provides Earth normal gravity to the passengers and cargo. For more about this see Space Travel Using Constant Acceleration.
As I mentioned in the beginning, I would model the entire system around the railway of the early 20th century. Through a series of transfers, a person could travel across the US with little fuss.
Constant acceleration even works for interstellar travel as well. It would take 1 year + the number of light years to reach any given star. So Alpha Centauri would take 5.2 years since it's 4.2 light years away.
While I'm writing this I thought of a possible propulsion system, an EM Drive or an RF resonant cavity thruster. I know next to nothing about the science of the drive, but my understanding is that it's a "fuel-less" drive system. An engine produces an EM field, which it needs to run the lights and whatnot anyway, it diverts a portion of the EM to the propulsion system and it produces thrust. Anyway, it's a thought. 947