Did they ever mention anything in the new canon about where storm troopers are trained?
We learned in TFA that Finn
was taken away as a small child and trained
In the Legends canon, they are trained on Cardia and then sent out, and clone troopers were trained on Kamino and other training facilities across the galaxy. Is there anything how both the First order and Imperial storm troopers are trained?
From reading the prequel book "Before the Awakening", which deals extensively with Finn's training and run-up to Jakku battle:
The name of his training location(s) is never mentioned
However (as detailed in my other answer on the site, with proof - or another, same-themed answer), at least a part of his training was on the Starkiller world, in the infamous "sanitation" scenario
The only 3 things he did AFTER his training finished was:
Embark on a ship to travel to his first post-training assignment (Pressy's Tumble mining center on an asteroid in Pressylla system)
Bungle up that assignment by failing to shoot at unarmed civilians
Get swept up with everyone else there by Captain Phasma, to go meet Kylo Ren and depart for Jakku
This means that, whenever he did his "sanitation" work on Starkiller, it was before his training ended.
However, we can theorize the training was 100% on Starkiller Base, from the fact that it was First Order’s main base as per Forster novelization:
“Finn’s familiar with the weapon that destroyed the Hosnian system,” Poe said. “He worked on the planet where it was built.”
Leia took Finn’s hand. “We’re desperate for anything you can tell us.”
“It’s located on the world that serves as the First Order’s main base,” Finn said. “I’m sure that’s where they’ve taken my friend. I need to get there, fast.”
There is an unconfirmed fan theory that Starkiller Base was the world known in prior canon as Ilum.
New Disney Canon:
One of the academies for Imperial training was on Arkanis.
Visual Dictionary page for General Hux says that his father was a "highly placed official in the Imperial Academy... on Arkanis" (later in the same page it said the father was the "Academy commandant")
As @thegreatjedi noted in a comment, we are told (The Visual Dictionary) that Galactic Concordance (the post-Jakku peace treaty between New Republic and remains of Empire) stipulated abandonment of Imperial Academies (plural) so there likely were more than one.
EU/Legends
"Star Wars: Imperial Handbook: A Commander's Guide" by Daniel Wallace (published during Disney Era, 2014/10, but explicitly designated as Legends), covered a list on pages 21-24, under "Military Recruitment" section. It listed:
Prefsbelt VI Naval Academy
Coruscani Pilot Institute
Raithal Academy (for Army officers)
Corulag Imperial Military Academy
The next are Strormtrooper academies:
Caridan Military Academy
Royal Guard Academy of Yinchorr
