You have to choose one ED to start with. You have to choose something from the sphere your class(es) place you in. Druid puts you in the primal sphere, so your choices are: Primal Avatar, Fury of the Wild or Shiradi Champion. Once you choose a destiny, you must level it to level 3 before you can switch to an other destiny. Note this is a switch, you only have one active destiny at a time.
Level 3 allows you to move to another destiny in the same sphere. There are also specific destinies in the destiny map that act as gateways to other spheres. Shirardi Champion for instance is the gateway from the Primal Sphere to the Melee sphere, so once you level Shirardi to level 4, you can move over into the melee sphere.
Your Experience points from quests also level your active destiny. You continue to level your destinies even after you reach level 25 and are capped from an epic level standpoint.
For each 3 levels you earn in any destiny, you also get 1 fate point. You use fate points to open up or increase the level of twist of fate slots. A twist allows you to borrow one ability that you have already earned in a non-active destiny. Twists are pretty useful, so in general you end up runnning up a lot of destinies just to earn fate points so that you can have more and higher level twist of fate.
Here is a site that lets you see the destiny map, and look at the features of any given destiny. Click on the Map tab to see how the destinys connect to each other, or the tree view to see what features are available in a particular destiny:
http://ddodestiny.info
Destinies level pretty fast, and you want to level several of them, so choosing the wrong one is a minor annoyance not a set back. You will want to eventually level all three of the primal destinies at least to level 3 or 4. If you are primarily melee, I'd start with Fury, if you are an active caster, give Primal a try. But you will be able to switch to another after you run a few epic dungeons so it doesn't make that much difference.