Current probes rely on the timing of planetary alignment to allow for multiple gravitational assist manoeuvres. The Voyager and Cassini probes used this to great effect.
With current technology, but far more funding than is currently available, we could construct a solar sail that is projected to reach 0.1c, or 10% of the speed of light. This would essentially be like a sail on a yacht, that uses light rather than wind. Light has momentum, even though it doesn’t have mass, so absorbing its energy would accelerate the sail.
Theoretically, the speed of light is the limit. There has been some exciting research in the field of propulsion recently that has given reason to think we may get close. But the techniques are barely at the experimental stage, and would still take many years to accelerate from an Earth orbit to anywhere near the speed of light.
The big problem with current propulsion systems is Newton’s Third Law of Motion. In order to accelerate the large mass of a rocket, we need to eject the smaller mass of burnt fuel at great speed. So we are limited by the mass of the fuel we can load onto the rocket, and the speed at which it can be ejected. If interstellar travel is to become a reality, we need a propulsion system that doesn’t eject matter, such as the solar sail.