From Cassini's launch until today, “gravity assists” have been an essential component of the process of making the spacecraft go where the scientists want it to go.
This propellant-saving, mission-enabling technique has been used in solar system exploration since the early 1970s. The key to the gravity assist technique is that it involves three bodies – the spacecraft, the solar system body that is providing the “assist”, e.g., Venus, Earth, Titan, and the central body about which the spacecraft’s path is being controlled. So early in Cassini’s flight, the three bodies were Cassini, Venus, and the sun. More recently, the three bodies have been Cassini, Titan, and Saturn. It is frequently asked how it can be that the gravity assist technique doesn’t violate the law of conservation of energy, since it might appear that energy is being gained for nothing. This fundamental law of physics is of course fully in effect during all such maneuvers. In the two-body problem, e.g., Cassini and Titan, there is an exchange between potential and kinetic energy as Cassini speeds up relative to Titan on approach, and then slows again to the same speed as on approach during departure. But the total energy is constant throughout this period, and Cassini leaves Titan with exactly the same energy relative to Titan as it had before the encounter. In the three-body problem, however, with Saturn at the center, the interaction between Titan and the spacecraft has caused an exchange of energy between the two bodies, so while the two bodies still have the same energy relative to each other, their individual energies relative to the central body have changed. Relative to Saturn, Cassini will have gained energy and Titan will have lost it, or vice versa, depending on how the flyby trajectory is designed, but the total system energy (Cassini, Titan, and Saturn) is unchanged. Since the energy of a body is determined by its mass and speed, and the masses aren’t changing in the gravity assist scenario, only the speed changes need be considered here. For a fixed energy change for Titan, only an infinitesimal speed change is needed since Titan is so much more massive than Cassini. Likewise, Cassini needs a rather large speed change to get a matching but opposite energy change since its mass is bordering on insignificant relative to that of Titan. A typical close Titan flyby will change Cassini’s speed by around 800 m/sec relative to Saturn (but zero relative to Titan), and such a speed change translates into substantial orbital parameter changes in Cassini’s orbit about Saturn. Titan, on the other hand, will experience a speed change of around 7 cm per million years, so no one need worry about any changes to the orbital parameters of Titan!
The rocket that launched Cassini in 1997 was the most powerful available to NASA, but it still wasn't powerful enough to send the nearly 6,000-kilogram (13,200-pound) spacecraft on a direct course to Saturn. Instead, mission designers planned multiple flybys of Venus, Earth and Jupiter, using each planet's gravity to boost Cassini's sun-relative speed and send the spacecraft out to Saturn.
Cassini's main onboard rocket engine was needed to brake the spacecraft and allow it to be captured into an orbit about Saturn on arrival in 2004. But since then, most of the “steering” of the spacecraft has been done through carefully designed flybys of Saturn's large moon Titan, which has more gravitational influence to offer than any of the other moons. Knowledge of Titan's mass and orbit allows mission planners to choose flyby conditions that will result in the desired amount of change in Cassini's direction and speed. Each Titan flyby is targeted to return Cassini to the next Titan flyby. Close flybys of other interesting moons are made along the way as the spacecraft continues to loop around Saturn.
Titan gravity assists have been used to achieve significant changes in the inclination of Cassini's orbit as well so that instead of staying nearly in the equatorial plane, the spacecraft's flight path has been inclined well out of the plane of the rings. This change in the viewing geometry has brought many new findings of previously unseen ring dynamics and atmospheric phenomena at Saturn's poles.