At level 25 you will be given a quest titled "Testing Your Mettle". This quest requires you to speak with Jucleas in Sanctum. Jucleas is the same person who performed your Daeva Ceremony and also the same person you had to talk to for your Stigma quest.
You will want to speak with Jucleas to see a quick cutscene. Afterwords update your quest. You now need to speak to Rinos. Rinos can be found directly east in the Coliseum.
In order to get to Rinos you will need to run through the Coliseum. To enter you will want to speak with the Arena Master, Epeios. Once inside Rinos can be found at the other end. This part can be quite annoying if people are in here fighting. After you talk to Rinos and update your quest you will need to return to Jucleas.
Speak with Jucleas and to turn in your quest. Accept the new one from him called "The Abyss Quiz". For this quest you will need to head to the most northern point in Sanctum, The Library of the Sages.
Here you will need to speak with Gromoros, He will quiz you on what you know about The Abyss. Below are the Answers to this Quiz.
After you finish your quest at Gromoros you will need to head a bit southeast to Nestor who has another test ready for you.
After you finish your quest at Nestor you will want to continue further north in the Library to Xenophon. Xenophon has your third and final test.
After you've completed your final test you will want to speak with Fuchsia and turn in your quest. Now Fuchsia will give you a new quest with 3 different options. The easiest option to pick here is "I will go to Eltnen" any other ones will force you to invest much more time then you should.
With new quest in hand you will want to go to Eltnen and speak with Telemachus, the General of Eltnen Fortress. Talk to him and refresh your quest. You will now need to kill 3 different types of beasts 3 times. First one you will want to kill is the Roaring Monitor which can be found around the Putrid Mire.
After you kill all 3 of the Roaring Monitors that you need to kill you will want to head south west to Eiron Desert. Here you will find Foraging Kurin.
After you kill all 3 of the Foraging Kurin that you need you will want to continue south to the Bighorn Baku area.
After you kill all 3 of the Bighorn Baku's that you need to kill you will want to return to Telemachus and turn in your quest. Telemachus will give you another quest that requires you to speak to Daedalus. Daedalus can be found just outside of Eltnen fortress on a floating island. Below is a video to better explain this quest.
After you complete this quest you will want to speak to with Daedalus again. He will redirect you to Telemachus. Talk to Telemachus to complete your Abyss Access Quest. Now you will have access to The Abyss. The easiest towns to get there from are Veteron and Heiron, the one in Eltnen is kind of in a bad place.
If you want to get to The Abyss from Eltnen you will need to fly under the tree to the Abyss Gate. The Abyss Gate is a large ring with a lot of blue coloring around it. When you get close enough and you mouse over the center of the ring it will pop up with a window letting you know if you right click it you will gain access.