Getting off the ground is not the problem. Accelerating to Orbital Velocity (~ 18K mph) or Escape Velocity (~ 25K mph) is what takes all the Energy/Fuel/$$$$$$$
BTW, when one of our kids was at Embry-Riddle Aeronautical University in Daytona back in the mid-1990s, they had a model of the Space Elevator in a glass case in one of the Administration buildings.