Since a lot of people are playing for the first time, or rerolling a new toon, I'd figure I'd try to put together a very basic overview for leveling efficiently. I'll try to go level by level, but if you are playing as a Free to Play Account, you won't be getting the subscriber XP bonus, so you may tend to be a tad behind the guide, so if that's the case, just grab some extra quests, run an extra level-appropriate flashpoint or two here and there, and play a few PvP matches and you'll be right on track. I highly recommend you do not skip the lore and story scenes, it's well worth it to watch them!
Some very minor spoilers may lie within.
Some general points that are very important:
Make sure you do the bonus quests for every quest you do that is level appropriate. They are often worth more than the quest itself and don't require deviating from the questing areas often, if ever. Make sure you log out in a rest area (most convenient is usually the fleet). Being rested grants you double experience for mob kills. Stop and kill any elite at your level or 2-3 above you, they are worth nearly as much as a minor quest and take less than 30 seconds to kill most of the time. Unless you are leveling as a group, I recommend that you quest in your AOE DPS spec (you can change this at any time on fleet, or purchase a character perk with a legacy level requirement and credits to allow you to change this at any time, anywhere). The AOE DPS specs are Rage for Marauders and Juggernauts, Madness for Sorcerers and Assassins, Lethality for Operatives (once you are able to get corrosive grenade), Engineering or Lethality for Snipers, Advanced Prototype for Powertechs, and Arsenal for Mercenaries. These specs allow you to kill most mob packs quickly, allowing you to complete "Kill X" and "Gather X" quests expediently. Complete your flashpoint and PvP dailies every day, without exception. They grant considerably more XP/time than planet questing, and you can queue for them from anywhere.
And now a very general overview of where you should be, by level:
Levels 1-10 are pretty straightforward. Follow your class quest, grab every quest you can find on your starter planet, and complete them. You will get a few heroics, that, if people don't happen to be around looking to group for them, can be very difficult. Skip them until you get your first companion, at which point even the Heroic 4 quests should be fairly easy. Once you complete your class story, you will be told to go to the Fleet. Before you go, make sure that you are either level 10 or within roughly 2000 experience of getting there.
Upon your arrival on the fleet, there will be some mission terminals that let you acquire your crew skills. Make sure to talk to each crew skill trainer, you get a large amount of XP for your level. Grab your first companion quest from your companion in the cantina for some more quick XP. At this point you should be, at least, level 10. Choose your advanced class, and hit the Priority terminal, Combat Training Terminal, and Supplies Terminal. Each of these will give you some introductory quests. Complete them, and the dailies they unlock. At this point you should be nearing level 13. If you are below level 13, do a few extra runs of Black Talon (there are almost always people looking to run the flashpoint!) until you are. You should leave the fleet for the first time between level 13 and 15.
Upon arrival on Dromund Kaas, you'll be given an option to take a speeder directly to your next class quest. I suggest that you take the long way and walk down the hill into the jungle, grabbing all the quests along the way. You'll gain at least a level or so, and get some map exploration that will help later. By the time you arrive in Dromund Kaas proper, you should be roughly level 15-16. Do the fetch quests around the city, and head to your class quest. The class quest from here will hand-hold you through the rest of the planet, which should wrap up around level 19 or 20.
Balmorra. There is a reason you are over-leveled compared to this planet's recommendation. Nearly everyone who has leveled through here more than once hates it, and rightly so. There is a lot of walking, and not a great quest density. If you are looking solely to level efficiently, you will want to simply do your class quests, ignore side quests, and get off of the planet as quickly as possible. You should have leveled enough on your time through the end of Balmorra, so make a pit stop on the fleet to grab some new gear with your planetary commendations and hit the Supplies terminal for a new Flashpoint daily. Queue for it while you continue onwards.
Nar Shaddaa. Arguably the coolest planet in SWTOR. Let the class quest hold your hand and bring you hub to hub. Do all the quests you can, especially the bonuses. Mob density is high, but they generally don't block travel paths, which makes Nar Shaddaa an excellent place to grind and over-level a little. Make sure you do as much as you can here, and if you overlevel a bit, that's a good thing. Once you're all done here, back to the fleet again, check if you can pick up any new dailies (or maybe it's a new day so you get a fresh set!), and spend some comms on some gear for yourself and your companion.
Note: After this point, the level you are can start to vary depending on how much rested XP you've been using, how many dailies you've done, and how many "out of the way" mobs you've killed. It shouldn't be too far off, but don't be alarmed if you're a bit ahead (skip some sidequests, or save your XP lead and keep questing hard for the upcoming full planet skip) or behind (grind a little, do some extra PvP or Flashpoints).
Tattooine! As Star Wars as Star Wars can get. If you're a lore lover, you'll have a field day here. Do the planetary quest (picked up in the spaceport), some side quests that you move past, the heroics if you can find a group, and, of course, your class quest. You should be just about level 30 or 31 when you finish all that, but before you leave, pick up the bonus series at the spaceport! It's a whole new quest chain with sidequests (every planet has a bonus series, but most require backtracking or are inefficient as far as XP/time). Do the chain and you should be roughly level 34.
Here's where you can get a little creative with your leveling. You're headed to Alderaan. You can choose to do just your class quest (much like Balmorra), you can do the class quest and main planetary story line, or you can do the class quest, planet quest, and some extra sidequests (make sure they're at least green in your quest log). This should leave you anywhere from level 33-35. Now is a really good time to spend all those planetary comms you have when you head back to fleet.
Taris. Ugh. Taris is the bane of efficient leveling. It's slow, mob dense, and not all that rewarding. However, the planet quest leads you to discover some AWESOME lore, so if you're interested in the story definitely do that. If not, grab your class quest, power through it, and get out. This is the main planet that power levelers skip.
Quesh. Quesh Quesh Quesh! This planet is awesome. It's got a great storyline, awesome design, and best of all, it's small. All the quests are near each other, the mobs are well placed, and everything is lovely. Do every single quest here. It's an absolute XP goldmine. You're now getting pretty up there in levels as well, so the BP/Foundry Flashpoint quest chain should be available. It's awesome lorewise and great XP. Do that as soon as you can.
Hoth. Land of the tauntauns. Let your class quest drag you hub-to-hub, but don't venture too far off the beaten path. There are opposite faction level 50+ zones and world bosses floating around, so be careful. Hoth is pretty easy to get around, quests are basic and rewarding, and once again, the design and storyline are really really great. Enjoy the nice XP gains, and head back to the fleet to re-gear and prepare for your next planet.
Belsavis. Lore, lore, lore, and more lore. My personal favorite planet. Do all the quests, especially the area ones that pop up as you move hub-to-hub. The XP gain here is really, REALLY good. Enjoy the story, and we're pulling into the home stretch of leveling. As an Imperial don't go all the way North, scary world boss + Republic daily area = sad lowbie Imp.
Voss. The quests here are long, but well worth your time. Do the planet quest (unless you end up outleveling it), your class story, and a few sidequests. This is a heavy world PvP planet, so be careful (or enjoy ganking the filthy Pubs).
If you are level 47 at this point, go pick up the quest "Crystal Ball" from an NPC on the fleet. This is a quest intended for the "old" elder game where the level cap was 50. By doing this full quest chain and the side quests (which become dailies), you should hit level 50.
If you are not level 47, or don't hit 50 at the end of the Crystal Ball quest, head to Corellia, and finish your class quest. If you did Ilum, you should be 50 or 51, and if not, you should be 47 to go do Ilum. You shouldn't need to do the planet or side quests, but they are pretty cool story-line wise, so don't skip them if you've been following the galactic story.
Now that you're level 50, the game changes quite a bit depending on what you have available. Use your planetary comms to get the best gear you can, and it's time to start prepping for the elder game at 55. The easiest way to do this is to go to Makeb (which has a lead-in quest on fleet and on your ship). Follow the Makeb storyline, grabbing the sidequests (which become dailies after you complete them once). At the end of Makeb, you should have a nice amount of basic commendations (which buy introductory level raiding gear) and be very near level 55. If you do hit 55 here, you're all done. Go run dailies to get some level 55 gear and be ready to raid!
If Makeb doesn't get you to 55, though, don't fret. The rest of the grind is really easy. You can get together with some friends or PUGs and do some Hard Mode Flashpoints (Level 50) or Level 50 Story Mode Operations. Between a few of those and a game or two of PvP (with the daily) you should ding 55!
Let me know if anything is unclear or you need specific advice about any of this at any point.