I have an idea of running a space-travelling game. I was thinking something like Shadowrun, only on an interstellar scale. What are some good systems I should look into using?
I'm looking to do something similar to the Firefly TV show or Serenity Movie (same universe), or maybe something like Star Wars only without Jedi/The Force.
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The Mongoose Publications flavor of Traveller is able to handle quite a bit of that. The core rules cover ships (to about 28,000 cubic meters in size... not really that big!), have a simplistic system for cargo availability (it's no good as a simulation, but it plays well and serves dramaturgically), has a psionics system that works well, has character and world generation systems, some cybernetics, and several drive options. Further, several supplements add more; Strontium Dog adds a magic system closer to that of SR in tone, The Psionics supplement adds a lot more to psionics as well as some more transhumanist elements, Merchant Prince and Scoundrel add some playable options for larger scale economic focused games. High Guard allows bigger ships.
Classic Traveller or Traveller: The New Era can both be used to do that type of game, using psionics as magic if desired. TNE can also be combined with Dark Conspiracy 1E/2E, a horror game using a variant of the same psionics rules but adding more magical options to it. CT, TNE, an DC are available from Far Future Enterprises on cd-roms.
WEG's d6 Space is generic, but since it's essentially what would have been D6 Star Wars Rules 3rd Ed, but stripped of overt Star Wars references, it clearly is capable of space opera. It lacks the trade system mechanics of d6 SW (from a Galaxy Guide), but can easily be used with that system if you can find the book (WEG is essentially out of business). d6S core is a free download.
WEG's Septimus is also d6 system, and is a space opera setting. Septimus is available free in PDF as well.
FGU's Space Opera is still available; it's rules-heavy, setting light, and predates the cyber craze, but it does have a wicked psionics system.
You might want to look into Starblazer Adventures or Diaspora, both using the Fate engine. Diaspora is more hard sci-fi, and while Starblazer is very much pulp-comic Sci-Fi. Both cover ships, traveling, and a variety of other issues, and both get excellent reviews. And the books for both are big, beautiful tomes.
Another possibility is to use any of the Unisystem games with the "unisystem Sci Fi Setting" in Eden Studios Presents Vol 1. The setting is explicitly a space western setting, and while focused on use with Cinematic Unisystem games, it could be very easily adapted to any of the Full Unisystem games.
Recently released is the Star Wars: Edge of the Empire, and the Edge of the Empire Beginner Game. Using custom dice, the game feels like a port of Firefly into the Star Wars universe in many ways. Characters are rogues, criminals, and desperate folk of the fringes, with a ship, and a motivation to keep it flying. Obligation rules make for interesting group dynamics, and sometimes even parties working at cross purposes. The custom dice may present a difficulty for some players and GM's. The beginner box includes a single set; separate dice sets are available for around US$15. Custom die-rollers are needed to generate the symbols used, and one was released for both iOS and Android (albeit $5).
On the secondhand only score...
Margaret Weis's Serenity RPG was the licensed game. The dice mechanic is less popular, there is no trade mechanic, no system generation, and no world generation, and ships are treated more as characters than as objects. It went out of print in 2010, and the PDFs pulled when the license ended in 2011. A new version, under a Firefly license, is being released in 2014, but a preview is being released at GenCon 2013 (18 August).
Alternity can do it well, too, but lots of people dislike its dice mechanic.