If the planet is 20 light years away, you're saying we 
  could reach it in less than a year if we attain a velocity
  of 0.9999c?

Yes. In the reference frame of the traveler, very little time passes. However, when he returns, people on earth will have aged more than 40 years.

This is also why particles traveling at c cannot possibly decay: no time passes for them. A photon is everywhere at once, from its own point if view.

Amazing. I hope to some day wake up and have a deep conceptual understanding of the relationship between velocity and time in relation to the various bodies at play.

I also hope to have an understanding of what it means for a photon to be everywhere at once, from its own point of view.

In the end, I guess I just want to understand the universe just a bit more than I do.

Special relativity is actually quite accessible and really interesting - find a good book or set of lectures on iTunes U or something. (General relativity, the generalised version including acceleration and gravity, is what makes people think relativity is hard).

  This is also why particles traveling at c cannot possibly
  decay: no time passes for them. A photon is everywhere at
  once, from its own point if view.

That's a really great explanation. For the first time, I've been able to visualize how light can be both a particle and a wave. Thank you.

Time dilation (special relativity) has nothing to do with wave-particle duality (quantum physics). The infinite time dilation of photons is due to their velocity, not their wave-like nature.

Furthermore, all particles (electrons, protons, quarks, etc.) have wave-particle duality just as much as photons do.

Wave-particle duality is even observable with larger molecules: http://link.aps.org/doi/10.1103/PhysRevLett.88.100404

Thanks for the correction. s/wave/continuous beam/.

According to the traveler you're traveling as fast as you think the traveler is traveling. The only way to make that work out is that to the traveler the destination is much closer than it is for you. This is called length contraction.

And rounding it out, different observers disagree on which events are simultaneous. In particular until the traveler turns around, the traveler thinks that the Earth was left recently. After the traveler turns around, in the new reference frame the traveler left the Earth close to 40 years prior.

Relativistic travel would be one-way essentially. It would allow one to travel the visible universe within a current human lifespan (of course, with ridiculous power requirements). Anything left behind would have experienced thousands of millions of years of time.

Sci-fi dealing with this topic: Forever War, Ender's series, a few stories from Niven's Known Space universe.

See also the anime Gunbuster, which has teenagers piloting giant robots (so far no surprise) into deep space at measurable percentages of c. Getting back after saving the world they find their school friends have grown up and had kids.

Right. Time/space dilation effectively makes the trip shorter for the person traveling near c.

Here's a really mind-blowing thing: If you were able to accelerate at a fairly reasonable rate indefinitely - say - using an interstellar ramjet, you could conceivably circumnavigate the entire universe within a human lifetime (in your own frame of reference, of course)

Of course, interstellar ramjets might not actually work in practice... but still, the concept of time/space dilation holds.

Last I heard from the hard science fiction people, such ramjets have an upper speed limit imposed by some sort of drag. Can't remember for certain... maybe Atomic Rockets has something on them... yeah, http://www.projectrho.com/rocket/slowerlight.php#Bussard_Ram... . They say 0.12c is the best speed for ramjets.