Note: Keep in mind that these methods may be subject to change at any time if a new patch implements changes.
Edit #1 (27/09): And just as expected, Bungie nerfed the spawn rate of the cave. Changing the spawn time from six seconds to forty. So this method is now far less effective. The author is currently looking into other profitable methods of farming and will update with the results shortly.
Edit #2 (02/10): Resourceful Destiny players have found an alternative farming location in the Rocketyard area of Earth. Details can be found below. Video Guide will be added after uploading (Around an hour from now)
Loot Cave 2.0
The current best place to grind is in the rocketyard area (to reach it you need to head up the stairs on the building behind you from your starting position one Earth, the opposite direction you would travel for the prior loot cave). Head through the structure and follow the trail outside until you reach the rocketyard area. Watch the video to see exactly where the "cave" is located (it's more like a door or hole in the wall) and the location you should stand for the best results.
The "cave" can be done solo, but is more efficient with more players as stragglers are less likely. Keep in mind the following things though.
All enemies in the local area must be killed before the cave will start spawning.
Not killing all the emerging enemies will pause the spawn until they are dealt with.
Being too close to the "cave" will stop the enemies spawning (useful when picking up the dropped items).
Efficiency, Weaponry and Levelling:
The most efficient weapon to use in killing the enemies in the cave is a scout rifle with a high impact rating. This determines the amount of damage done by headshots. Having the right weapon means you can one shot the enemies, even when you are at a low level. It also prevents wasted ammo and will improve your skills in head shot aiming.
Most players will wait until their ammo is depleted to collect items and then rush the cave, killing the spawning enemies (preventing them running away and having to be hunted down afterwards). You will gain a constant stream of experience by grinding in this area as well. It is possible to reach level eighteen from seven in a single night (I've done it personally). Being in a grinding group also relieves the monotony of the activity.
For those players who want to level the more "legitamate" way. Completing story missions will give a chunk of experience, along with bounties also giving large amounts. Doing strike missions with high enemy numbers can also work well.
Loot Cave 1.0
Previously the best location to grind for items (and also experience) could be found on Earth. From the starting location on Earth patrol, you need to head to and then through Skywatch (the large building with the radar dish on top). Head back outside and look for the large green spires surrounding a crashed hive ship as you emerge. Next to the crashed hive ship is a small cave which spawns a continuous stream of three acolytes and then three acolytes and three thralls.
Below is a video showing how to reach the destination and how to farm items when you arrive.
Once at the location you will need to keep in mind a few things:
Being too close to the cave will stop the enemies spawning (useful when picking up the dropped items).
Not killing all the emerging hive enemies will pause the spawn until they are dealt with.
Sometimes after remaining in the area for a long time, both enemy groups in the area will join forces and attack all guardians in the area. This is announced by text in the bottom left of the screen. Several waves of enemies of increasing strength will appear and continue to attack for a set period of time. They will dissappear after a set amount of time and things will return to normal. If you are the only guardian in the area, a simple death and respawn will reset the area. Otherwise you will have to wait for the timer to expire.