I accepted the quest [level 12] to collect Dryad tears and I have the green necklace in my quest inventory, but I've killed several dryads and nothing is registering. Have I missed a step/element?
Quest items don't drop 100%; i.e. killing a mob doesn't mean you will get the quest item. Sometimes, you need to kill dozens before you get the item. For some painful quests, you need to kill hundreds before you get the items you need. - -; But yeah, just keep at it xP
Quest item drop rate varies by quest. Some give a 100% drop rate...others can be less than 1%.
I'd suggest a few quests to try...but nothing really stands out for me from my time in the LE starting town. There are the weapon quests, but I can't remember which those were. (L1-L20 went FAST for my LE....but then again, since she was my last sub-20 char I had top gear for her anyway.)
Once you head out I have a few quests that I always keep active on my manor char (perma L22-32), but those are all L21+.
Just keep at the quest, eventually you'll get the drops.
Also, make sure you are killing Dryads and Dryad Elders.
Green Dryads do not drop the quest item.