I'm building a Warlock with an Archfey patron, and I want to read more about the Feywild. I remember the AD&D Manual of the Planes and it had information from all the planes. Does 5e have one of these? If not, is there a place I can read in-depth about the Archfey and the Feywild? (Similar information on Hell, demon princes, and the Abyss would also be greatly appreciated.)
The Dungeon Master's Guide and Mordenkainen's Tome of Foes are your best bet
The DMG has a chapter on the planes, which does include the Feywild on pp. 49-50 (and the Nine Hells/Abyss too, pp. 64-66 and 62-63 respectively).
There is more info in MToF on both the Feywild and the Eladrin on pp. 49-50 (what a strange coincidence!), as well as an entire chapter on the Blood War, with plenty of information on devils, demons, the Nine Hells and the Abyss (pp. 5-34).
There is also a tiny bit of information on a few Archfey in the Sword Coast Adventure's Guide, p. 138, but it's basically just a sentence or two about half a dozen or so possible warlock patrons. Not really worth being a proper part of this answer, but just thought I'd include it for completeness.
No. At this time, there is no equivalent to the Manual of the Planes.
However, Modenkainen's Tome of Foes does contain info about demons and devils with regards to The Blood War. There is also info about the Feywild and it's occupants. In my opinion, not enough.
Some books from earlier editions of D&D might be useful
A lot of the lore from sourcebooks for earlier editions of D&D may still be useful for D&D 5th edition campaign. Much of this lore has been retained in current D&D canon, though it is explored more thoroughly in the older sourcebooks.
The D&D 4th edition Manual of the Planes (2008) has 16 pages on the Feywild and the major archfey and important realms. The Feywild is largely an invention of D&D 4th edition, though it is synonymous with the Plane of Faerie, which was mentioned only briefly in earlier sourcebooks. The 5e lore on the Feywild is largely the same as 4e, with minor changes to things like the name of Queen Titania (Tiandra).
For the realms of demons and devils, the absolute best sources are the D&D 3rd edition Fiendish Codex I and Fiendish Codex II. The 3rd edition Manual of the Planes and Book of Vile Darkness also have information on these topics, as do numerous AD&D Planescape sourcebooks including Hellbound.
Dragon magazine also ran a Demonomicon of Iggwilv article series detailing specific demon princes in issues 329, 333, 337, 341, 345, 349, 353, 357, and 359; in the digital Dragon issues 360, 364, 369, 376; and in the digital Dungeon issues 172, 188, 205, and 208.
The Codex of Betrayal series similarly detailed archdevils, appearing in digital Dragon issues 365, 373, 427, and 428; and digital Dungeon issues 176 and 197. If I'm not mistaken, you can still pay for a single month of the 4e D&D Insider service and download the entire back-catalogue of Dragon 360 onward and Dungeon 151 onward.
The Court of Stars article series also describes powerful archfey. It appears in digital Dragon issues 374, 420, 422 and 428; and digital Dungeon issues 185, 196, and 205.
The Planescape Index can also help you to find AD&D 2e and D&D 3e sources for specifics on planar topics.