Wanna see the universe? Don't care about having anyone you knew back on Earth still alive when and if you return (perhaps not about getting squished to a pulp either)? Then special relativity (SR) is your friend. If you get your spaceship moving close to the speed of light, C (to keep calculations simple, suppose essentially instantaneous accel. and deccel. to constant velocities) then you can travel to a star system x light years away from Earth in ~x years according to a clock on Earth.
From Earth's frame of reference nothing much turns on whether your ship's cruising velocity is .99C or .99999C (cf., particle collision velocities in Cern's LHC are projected to reach .99999999C): either way you're taking just a little over x years according to clocks on Earth to get to wherever it is you're going. To you on the ship, however, how close your velocity is to C matters immensely: at .99C you spend ~x/7 years in transit, whereas at .99999C you spend ~x/223 years. If your destination is in the Alpha Centauri system (where Avatar's Pandora supposedly lives), then either way your outbound flight takes 4.4 years according to clocks on Earth. At .99C, however, your ship flight-time is just over 6 months, whereas at .99999C, your flight lasts a week.
Here's a graph for some intermediate ship velocities, focusing on in-ship flight times of whole years.
those possibilities require various sorts of relativity-breaching physics. But even if there's no way around SR, if getting very close to C turns out to be technologically feasible and survivable, etc. then humans could still spread out through the galaxy, but all but the very shortest trips would be importantly 'one way'. From Earth's frame of reference a relativistic diaspora would tend to happen on supra-civilizational or even quasi-geological time-scales. For example, a five year flight out for you at .99999C traverses 1000 light years and sees 1000 years pass on Earth. That's some frontier experience you've bought yourself, pard'ner.
Very close to C travel by a massy body like a space-ship might depend upon a particular kind of technological and physical breakthrough: essentially on finding some way to 'hide' or 'encode' a lot of the ship's mass as patterns of e-m. radiation. The limit case for this hypothetical possibility would hide or encode all of the mass. Complete masslessness, i.e., complete, (at least) temporary transfiguration of the space-ship and all its contents into e-m. radiation would allow and require travel at C, i.e., as e-m. radiation/light of some kind. That transfigured, massless, informational carrier state would, of course, involve no reference frame/point of view/time factor whatsoever. Decoding/de-beaming/re-materializing a ship with sub-C mass and time in a reference frame as the terminal stage of a 'flight' would be some trick! While realizing this possibility would require huge physical and technological advances, the possibility itself is relatively clear. It represents the least upper bound, limit case for diaspora through the universe that's compatible with SR.
It's somewhat heartening that the mass-less, travel-at-C limit case, while not the 'hyperdrive' of sci-fi dreams nonetheless allows for quite wild arrangements: people in ships get light-like encoded and 'beamed' to new worlds thousands of light years away. Civilizations rise and fall back on Earth while those light-like signals travel to their destinations, all before the next eye-blink of de-beamed/re-materialized time occurs for the people in the ships subject to this process. Their lives restart/continue without missing a beat, notwithstanding that that next beat occurs thousands of light years away, and thousands of years into the future according to clocks on Earth:
'I was just...' - Zap! - 'saying, that would be one small step for a man, and holy mackerel it was! Here I am completing my sentence in 2999 with Altair 4 hanging below me and off to starboard...'Star Trek or Forbidden Planet, but it's a hell of a space western model. One-way voyages of discovery and settlement into the far away and future: every ship its own, effectively linguistically isolated civilization (effectively the model of 1967's Planet of the Apes).
Two usefully bleak discussions of the feasibility of non-faster-than-light interstellar travel and associated colonization possibilities:
 While we effectively assume away accel. and deccel. to ease calculation, it's important that they're still 'really' there. If observers on the ship didn't accel./deccel. then both observers on Earth and observers on the ships would be moving inertially. Einstein's Principle of Relativity would then force all observations to be symmetrical, e.g., if one set of observers sees the others' clocks as running slowly, the other observers must see the first groups's clocks similarly slowed (i.e., since there would then be no frame-independent fact that the ship speeds away from the earth rather than the earth speeds away from the ship). Our cases require clock asymmetries, which implies that they essentially involve accel./deccel.. See John Norton's Einstein for Everyone course notes for patient, detailed explanations of the temptations and pseudo-paradoxes in this area. Wikipedia's time dilation page is excellent, but is too technical and gravitationally-oriented for non-specialists. Jesus was supposedly 33 when he was crucified. If, instead of dying, Jesus
- hopped a .99999C flight home to a planet 1000 light years away
- spent several years there generally recuperating and chilling out with descendants of his family
- hopped a .99999C flight back to Earth