Just as a word of insight - in order for the backstory to make sense, it's best to play the previous Trails titles:
Trails in the Sky 1st (recently remade)
Trails in the Sky 2nd (remake coming out soon, censored)
Trails in the Sky 3rd (not yet remade)
(these three known as the Liberl trilogy)
Trails from Zero
Trails to Azure (both part of an upcoming compilation)
(known as the Crossbell duology)
Trails of Cold Steel (the first of the four-part Erebonia saga) starts roughly 1/3 of the way into the Zero story and extends to the halfway mark of Azure, with Cold Steel II's halfway mark happening at the same time as the ending of Azure. The special epilogue of Cold Steel II is the first direct crossover mark between the Cold Steel series and the Crossbell duology and the end to the major storyline of the first half of the Erebonian saga). I personally suggest playing the Crossbell titles before the Cold Steel titles so aspects of the Cold Steel games make sense.
Trails of Cold Steel III starts the beginning of the wider Erebonian saga and how it affects the rest of the continent of Zemuria (where the series takes place), with Cold Steel IV leading to a mega-MASSIVE crossover of the entire Trails saga that left my poor PS4 wheezing, and the "epilogue" to the series 'thus far' (Reverie) both expands on the ending to Cold Steel IV and answering one major left-over question of the series while creating several entirely new ones thanks to a fake-out ending that's canon from Cold Steel IV (that gets semi-answered in Reverie & Beyond the Horizon, the third part of the Calvard saga - the fourth part has yet to be released).
Between the Crossbell duology and the Erebonian saga is a spin-off game, Legend of Nayuta: Boundless Trails. Fans..."debate" just how/if this game is even connected to the rest of the series. I personally suspect it does, else the developers wouldn't include a message at the end of Azure regarding the title. There are also certain aspects of the story of Nayuta that could fit in the wider series, especially certain revelations in the Calvard saga (Trails Through Daybreak I & II, Beyond the Horizon).
This last part is not a spoiler, but a strong recommendation should you play the entire series: Throughout the entire series you can collect storybooks that can be traded-in for special items. Not only should you collect every book, but I strongly recommend reading each book. The characters in the books? You will meet them, some will "join your party" even, and these books will help explain these characters in ways that the games themselves don't. Oh, and ignore the online discourse if you can. Daybreak & Daybreak II have certain story aspects that players believed broke canon...but Beyond the Horizon rather shut them all up quite firmly. Which means, to get the full story, you really ought to complete the various "optional dungeons" Reverie and Beyond offer.
The Trails series has some of the best writing you'll find in modern JRPGs, which isn't a shock as they're made by Falcom, one of the oldest names in the business. Dragon Slayer (including Legacy of the Wizard)? Falcom. The Legend of Heroes? Falcom. Xanadu/Faxanadu? Falcom. Ys? Falcom. Sorcerian? Brandish? Popful Mail? Gurumin? Falcom. They are what Square used to be.