Lunar Space Elevator
A lunar space elevator extends Earthward below the Earth-Moon L1 Lagrange point. A plausible lower terminus is the apogee of an Earth reentry orbit, with a perigee of (say) 6478 km. At orbit apogee, the angular velocity might match the Moon's angular velocity, a 27.321661 day or 2.3605915e6 second orbit, omega = 2.6616995e-6 radians per second. That is the apogee angular velocity of an Earth orbit with a perigee of 6478 km (skimming the atmosphere for re-entry, or leaving a launch loop at that altitude) and an apogee of 162,717 km. The average lunar distance is 384,400 km, the lunar equatorial radius is 1740 km, so the orbit apogee is about 220,000 km from the lunar surface. At 100 m/s or 360 km/h, that is 610 hours of climb, or 25 days. The semimajor axis of the transfer orbit is 84598 km is an additional 34 hours, for a total of 644 hours or about 27 days. That does not include lunar surface travel time.
A transfer orbit directly from a launch loop near the surface of the Moon might take 119 hours, or 5 days. There can be launch loops near every lunar outpost; For 3 gees to 2.38 km/s lunar escape velocity, they would be 95 km long.
A lunar space elevator would be horrendously expensive to deliver and deploy. A launch loop won't be cheap, but it will have much higher throughput (80 second launch cadence, 1080 vehicles per day, 27,000 times the payload rate) and much lower mass (the Moon itself carries the tension, only an energy-storing rotor is needed). Launch loop logistics will be much simpler.
No, we do not need a lunar space elevator.