Earth-Moon Lagrange Point 2 is accessed from an LEO Parking Orbit (100 nautical miles) for just 3.4741 km/s delta vee. Once there, Tankers from the moon can replenish the fuel tanks of a Starship. With a dry mass of 120 tons and payload of 150 tons, 3 Tankers launched from Earth are needed to do the task. A stripped back Tanker, massing ~100 tons, launching from the Moon would aerobrake to arrive in LEO and load up the Starship (120 tons + 150 tons payload) for the trip to EML-2. Two Tanker loads would be needed – whereas 3 would be needed from Earth.
Once parked at EML-2 Starship Tankers from the Moon can deliver 400 tons of propellant each, allowing a quick refuel. Tankers delivering to a vehicle in HEEO Staging Orbit would be far more limited, requiring 4 launches per 150 ton delivery. A full tank in HEEO would need 32 launches to deliver 8 lots of 150 tons each. Only 3 are needed to EML-2 if supplied from the Moon.
Source document: The utilization of halo orbits in advanced lunar operations
Delta-vee from Moon’s surface to EML-2 is 2,789 m/s descent phase (empty Tug return to refill). Ascent phase 2,667 m/s.
The Starship would then ease into an Oberth Maneuver orbit, dipping very close to the Earth for maximum effect. A Starship acting as a Space-tug might be a better option, to maximise the effectiveness of the full tank of propellant on the Starship. Some 6.55 km/s delta-vee. A Starship to Titan would need one tank load, for a 2 year flight, but a Starship to Callisto would need multiple staged launches, to carry sufficient propellant to brake directly onto the surface. Total trip time to Callisto would be ~600-700 days.