|MadSci Network: Engineering|
To answer the question in your subject line, the fastest speed that anything in the Universe can travel is the speed of light (186,000 miles/second). For objects with mass, this speed represents an unachievable limit, because it would take in infinite amount of energy to accelerate something with mass to the speed of light. These are all predictions of the theory of relativity.
There have been authors who have discussed the possibilities of "warp drives" and "hyperspace" (ex: The Physics of Star Trek by Lawrence M. Krauss). But they also agree that to make these things work requires the existence of matter or dimesions of spacetime for which we have no evidence of existence.
Your idea of using gravitational boosting from stars or planets is currently used to send spacecraft to other planets. However, to reach distant stars in reasonable amounts of time far greater velocities must be achieved than are possible using this method.
There have been many authors who have written on possible engineering designs to make interstellar travel technologically feasible. Journeys to distant stars might take many years for Earth observers, but for occupants of a spacecraft on such a journey the travel time could be made to be quite short if speeds reaching siginifcant fractions of the speed of light are managed.
If a spacecraft can reach such fast speeds relativistic effects will cause time to appear to pass more slowly for the spacecraft occupants than for those left behind on Earth. Reaching such speeds is a challenge to be sure.
Some have suggest the idea of accelerating at a constant rate for half of the journey (say at 9.8 m/s/s, which is the accelration due to gravity at Earth's surface). This can simulate gravity aboard the spaceship. At the halfway point the ship can turn around and begin decelerating at the same rate (maintaining the artificial gravity aboard the ship). Doing this can allow interstellar voyagers to reach neighboring stars in a matter of years, not centuries or even decades. The biggest challenge is that of finding enough energy to maintain the acceleration/deceleration.
There have been many suggestions for fuel to be used in interstellar journeys: fuel for chemical burning, for nuclear fission, or nuclear fusion, or antimatter for matter/anitmatter reactions. One intriguing idea has been to use a gargantuan magnetic field to scoop up interstellar matter like a ramjet and use it as fuel for fusion or antimatter reactions.
I am aware of one very good book that discusses many of these possibilities:The Starflight Handbook : A Pioneer's Guide to Interstellar Travel by Eugene F. Mallove & Gregory L. Matloff.
I hope this answer gives you a good start.
Bryan Méndez, UC Berkeley Space Sciences Lab
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