Entering and Exiting the Feywild

Entering the Feywild:
For the most part, unless you're using some pretty powerful teleportation magic, everybody entering the Feywild has to pass through the gates. There are a lot of different ways to get there - magic ponds, holes in the ground, mushroom circles - but they all lead to the gate. The gates are guarded by two archfey - Obie, a satyr, and Henry, an owlbear. Obie definitely thinks he's in charge, but that's questionable. He's very hyperactive and a bit much at first, but become his friend and you're set. Henry is kind of grumpy sometimes but he's cool too.
If you're with a Feywild resident, they should be able to vouch for you and get you through. He might ask to see documentation - I would just scribble your name and picture on a piece of paper beforehand and add a bullshit message about how you have permission to enter. If a Feywild resident isn't with you, that's when things get tricky and you have to pass a trial.
For as long as I've been alive, the trial has been the same - get one of the berries that Henry hoards. If you try to steal them or fight Henry, you'll get mauled in an instant. If you ask nicely, he'll tell you he'll give you one in exchange for killing one of the deer in the area. But, if you kill the deer you'll also fail the test and be punished. I think the idea is that if you're that willing to kill an innocent creature for your own gain, you aren't worthy of entrance. The solution to the problem is to somehow persuade or bribe or trick Henry into giving you the berries for something else. The Feywild is a land of twisted words and false diplomacy, so you must emulate that to get in.
Once you pass the trial, just give Obie your destination and hope. They do their best with the teleportation, but it isn't always on the nose.
Exiting the Feywild:
This one is a lot easier. Again, unless you have access to powerful teleportation, you have to go through Obie. Just yell out his name and say that you want to go home, and hopefully he'll show up and retrieve you. If he gave you a rock when you first came in, all the better - this helps focus the message. Once again, just give him your destination and hope. He's gotten decent at finding the camp by now, but if you want to go anywhere else you might be out of luck.