The fundamental shift introduced by Einstein in his 1905 paper on special relativity is that the idea promoted by Newton that space and and time appear the same to all observers leads to paradoxes. The rather more complicated idea that the shape of space and rate at which time passes changes for observers in relative motion is the solution he presented. The experimental verifications of Einstein's theory are so prevalent that there is no controversy about it in the scientific community.
It is often lamented that Einstein's theory (which includes the consequence that accelerating to a speed greater than the speed of light appears physically impossible) is a sort of spoil sport for the idea of human exploration of the universe. That conclusion is not true but there are remarkable consequences for those possibilities.
Assuming one could obtain a method to achieve relativistic speeds (i.e. speeds at some significant fraction of the speed of light) the combination of lorentz contraction and time dilation implies one could travel incredible distances during the normal span of a single human lifetime. Even more interesting is that if it were a round trip that human scale time frame of about 50 years for the traveler would contrast with the hundreds of million or billions of years that would elapse on Earth. This is the so-called twin paradox in spades.
So if relativistic speeds were achievable then it would be possible to travel quite literally into the far future. Don't make immediate plans because although the scenario only requires acceleration at 1 G (what we normally experience due to gravity on Earth, or 9.81 m/s2, or 35.32 km/h2), there are some immense challenges trying to find an adequate propulsion source, and there is daunting issue of shielding from relativistically enhanced collisions with the not quite empty vacuum and even microwave background radiation.
A derivation of the relevant equations can be found in chapter 6 of Gravitation by Misner, Thorne, and Wheeler or even more convenient is the link: http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html