The Battle Tree. Everyone has heard of it and many have already gotten to it. However if you are a little stuck on how to get to this new battling arena, I hope I can be of assistance. Firstly you need to complete the main storyline and have access to Poni Island (once again easily accessible through the main story).
I say you need to complete the story because you will not be able to make it through the gauntlet of trainers to get to the Battle Tree if you are not sufficiently levelled (around level 60 should serve you well).
You first need to overcome a specific Trainer who has helped you in relation to Zygarde, and who was first encountered in the Kalos Region. Known as Dexio, he challenges you to a non-optional battle. All of his Pokemon are above level 60 so be prepared for that. One of the interesting things about this battle is that he showcases and utilises Mega Evolution and as a prize for beating him, provides you with an Alakazite. This suggests that there may be more Mega Stones to be hunted down in the future of Pokemon Sun and Pokemon Moon.
You will not be able to proceed to the Battle Tree without overcoming Dexio so if you are having some trouble, you may need to do some training, or hunt down the Guardian Pokemon, as they are an easy source of power in the late game.
This battle will take place as you travel along the Ancient Poni Path on Poni Island. Continue travelling towards the Ruins of Hope (towards the right of the above photo) and as you get to the series of broken walls in the below photo, begin to make your way through them.
Travel through the next area and continue on through the gauntlet of tough Pokemon Trainers and powerful wild Pokemon until you eventually arrive at the Battle Tree and your new experience in Pokemon Sun and Pokemon Moon.