Getting off the Earth:
Earth's surface-satellite velocity is about 7.9 km/s. To get from the Earth's surface to low Earth orbit, about 300 km altitude, takes about 7.9 km/s of delta-V. Furthermore, it must be done with a high-thrust rocket, and that means a multistage chemical one. The Falcon 9 and similar rockets won't be put out of business anytime soon.
From
Falcon 9 - Wikipedia, version 1.1 has a fully-loaded mass is 505.846 metric tons and its LEO and GTO payload limits are 13.15 mt and 4.85 mt. Both its stages use kerosene-oxygen propellant.
Earth - Saturn:
The orbit with lowest delta-V is a Hohmann transfer orbit. One needs about 10.3 km/s delta-V near the Earth and 5.4 km/s delta-V near Saturn.
If one has a rocket with high enough thrust, one can leave the Earth fast enough to be at the right speed at interplanetary distances, with the help of the
Oberth effect as it's called. One needs a delta-V of only 7.3 km/s, but one has to apply it in a half hour or less. For a hydrogen-oxygen rocket, that means a mass ratio of 5.25, and that's propellant only. That means that the empty escape stage + payload can weigh at most 2.5 mt if launched by a Falcon 9.
For a low-thrust engine, it's worse. One's delta-V is 18.0 km/s
Formulas:
High-thrust: sqrt( (escape velocity)^2 + (final velocity)^2 ) - (orbital velocity)
(escape velocity) = sqrt(2) * (orbital velocity)
Low-thrust: (orbital velocity) + (final velocity)
It will take about 6 years to make the trip.
At Saturn, we need to get to Titan's orbit distance, about 1,221,000 km from Saturn on average. Titan orbits Saturn at about 5.57 km/s with a period of 15.945 Earth days. So if an engine can deliver the necessary delta-V in a day or two, it should be enough. Delta-V's:
High thrust: 4.0 km/s
Low thrust: 11.0 km/s
Next to get into Titan's orbital plane. It's close to Saturn's equatorial plane, which is inclined about 26.73 degrees to its orbital plane. One will need about 2.6 km/s of delta-V to do so.
Formulas:
High thrust: vorb * 2 * sin(inc/2)
Low thrust: vorb * inc (in radians)
Finally, Titan itself. Its radius is about 2575 km, but the safest altitude for a low orbit is likely 1000 km. At that orbit, its orbital velocity is 1.6 km, its escape velocity is 2.2 km/s, and its period is 2.8 hours.
High thrust: 0.6 km/s
Low thrust: 1.6 km/s
To de-orbit takes only 0.1 km/s, though one will crash into Titan's atmosphere at something like 1.5 - 2 km/s. That's a standard mode of re-entry for Earth spacecraft, however, so it's a pretty much solved engineering problem. One could do that while arriving at Saturn, though one will crash faster, at 10 km/s or so.
To get back into that orbit from Titan's surface will require 2.1 km/s, calculated by doing a Hohmann transfer orbit from Titan's surface. Once one leaves Saturn, one can arrive at the Earth by crashing into its atmosphere at about 18 km/s.