Okay it was pointed out to me that they changed the 4th boss in Eternity Vault to be more of a gear check (they require each raid member to defeat their own enemy by him/herself).
I have prepared the below list of pointers for newly-50's to gear up the fastest way. I hope it helps.
I posted it on the Zero Zero Zero guild forums, but I figured it might be useful on here as well. Feel free to post this elsewhere, but please link back here, and help me out with a thread bump
1.) Corellian Commendations
Finish Corellia, or at least do enough to *finish your class quest*. The quests for Corellia give Corellia Commendations (CoCs), and so do normal mode flashpoints at level 50. You can buy level 50 blue armorings/mods/enhancements/barrels/hilts or level 50 mod-able gloves/chestpieces/weapons. The fastest way to gear up would be to purchase the gloves or chestpiece if you don't already mod-able ones equipped or in the bank; I wouldn't recommend spending the CoCs on modifications because purple level 56 modifications are very easy to obtain. If you don't need any of the gear, I recommend saving up for the Ancient Artifact Storage Box for 36 CoCs. Inside is a random Level 50 Purple piece of armor. While there is no guarantee that you'll be able to use the item, they are BOE so you can sell the item on the GTN and use the profits to buy something you *can* use.
2.) Ilum PVE Daily Quests
After completing your class quest, go to Ilum and complete the PVE side of the planet (mobs hit hard, but you should be able to solo everything on that side, with the right companion and smart use of CC). It only takes maybe 1.5-2 hours to complete the PVE quest series. This unlocks daily quests for Daily Commendations (DaCs). There is a "mission support vendor" inside the republic base that sells level 56 purple Armorings for 8 DaCs. There is also a 2+ man heroic daily quest (difficult, ask the guild for help if you can't find anyone by asking general chat and I'd recommend a full group of 4, but I have managed to duo it once or twice) that rewards a level 56 purple Enhancement and 3 DaCs.
If you are diligent about completing the Ilum PVE dailies, you can get 1 epic Armoring and 1 epic Enhancement EVERY DAY.
If your also feeling particularly thick-skinned, you can take a beating at the PVP side of Ilum for the Ilum PVP Daily/Weekly and Champion PVP Gear Bags(more on this later).
3.) Belsavis PVE Daily Quests
Pick up the Belsavis Bonus series starter quest from the Fleet. By going threw this series, more daily quests are opened on Belsavis, which also award Daily Commendations. Like on Ilum, there is a daily commendation vendor on Belsavis. It looks like they have the same items for sale. There are also a few Heroic 4-man quests on Belsavis. One rewards a level 56 purple Mod and 3 DaCs. One rewards a level 56 purple Armoring and 3 DaC's. And another one rewards 3 DaCs and a level 50 level 50 blue version of companion weapons (best weapons obtainable
for companions that use techblades, techstaffs, or other unique-type weapons).
If you are diligent about completing the Belsavis PVE dailies, you can get 1-2 epic Armoring and 1 epic Mod, and a Best-in-slot companion weapon EVERY DAY.
4.) PVP!!! Warzone and Ilum PVP Daily/Weekly Quests
Yes, PVP gear is an excellent way to gear up. Champion PVP Gear bags are currently MUCH faster than grinding heroic-mode flashpoints for equivalent gear. In fact, even with the expertise stat on the PVP gear taking up itemization, the Champion pieces are BETTER THAN the gear found in normal mode operations (normal mode is considered "tier 1" and its the same as rarer drops in Hardmode flashpoints, hardmode is "tier 2" and nightmare mode is "tier 3". Nightmare mode is equivalent to battlemaster gear).
Once you hit 50, a "complete 1 warzone" quest becomes available from a terminal in the Combat Training wing of the Fleet. Easy. Once that one is turned in, the daily/weekly PVP quests become available. The daily is "win 3 warzones" and rewards 1 Champion Gear Bag, which contain 3 centurion commendations (used to buy centurion, or "tier 1" pvp gear) and a CHANCE at a token for Champion gear ("tier 2"). The weekly is "win 9 warzones", which is essentially complete the daily 3 times, but all 9 can be completed in a single day if desired/needed; the weekly rewards 3 Champion bags!
Another set of daily/weeklys becomes available at level 50. The Ilum PVP daily for "kill 30 imps" and the weekly "kill 150 imps". Unless you are very thick-skinned or just REALLY want the PVP gear as fast as possible, I would highly recommend staying away from Ilum until Bioware fixes the zone by making both factions numbers equal (and fixing the framerates). If you complete them, its another 1 bag for the daily and 3 bags for the weekly!
Warzones also give out Champion bags, but at a slower, and indirect manor. PVP Vendors inside warzones and on the Fleet sell mercenary commendations for warzone commendations at a 10 for 30 ratio. You can buy Champion bags for 200 mercenary +200 warzone commendations. So save up 800 warzone commendations, convert 600 of them into mercenaries, then buy the Champion bag. BE CAREFUL, when saving up to 800, because there is a hard cap of warzone commendations at 1000 and you don't want to hit it and lose out on any over that cap that you would have gained.
If you are diligent about completing the PVP dailies/weekly's, you can get 2-3+ bags EVERY DAY, and 9-22+ bags (and if your lucky, that could be around 4-5 pieces of "tier 2") EVERY WEEK.
Unfortunately, the highest levels of crafted gear is slightly lower than the highest levels of gear found through the dailies. By slightly, I mean by 3-4 stat points total. So if your a crafter, you can probably make yourself some really nice gear by reverse engineering greens to blues, and blues to purples. I wouldn't rely on this though, because unfortunately, its not reliable which patterns you'll get and due to the ease of dailies makes this sort of a waste of resources that could be sold or put to better use. The "Rakata" gear is made from Alloy, which is only dropped from final bosses in hardmode flashpoints. Crafting gear is an option.
While the datacron hunt might be pretty time-consuming and not seem worth it at first (I still haven't bothered to find any... maybe I'll do it this weekend), all those +4's can add up to some pretty nice numbers considering there are 3-5 datacrons per planet. All the stats could probably add up to the equivalent of an extra chestpiece, or close to one. In addition, matrix shards can be combined to make a level 50 relic with some really nice stats.
If you are dedicated about finding all the datacrons, they could potentially add up to some series stat bonuses, and a pretty nice relic. I would wait on this, it could be very time consuming.
There you have it, our fresh level 50 newbies! The best ways to gear up!
And its FAST too! Between doing dailies and PVP, I'd expect a new 50 to be geared enough for Eternity Vault in 5-6 days of doing dailies and modding out old armor in epics.
EDIT: Thanks to the datacron suggestion by CodeNameOrca!