I finally got to reading that chapter of the DMG, it's better than I expected for a short primer on the planes. There's some attempts to try to satisfy a lot of the player base, they even devoted a grand total of 1 short paragraph about the controversial World Axis cosmology. The chapter is very much about the Great Wheel and it reclaiming its "core" position in D&D. I'm certainly used to their being of course more details about the planes, having read through books on the subject.
Each of the Outer Planes gets at least a paragraph and an optional rule. The Abyss and Baator/Nine Hells of course gets the most details. Some of the optional rules on different outer planes are ones we are used to like Pandemonium's Madness and Limbo's chaos-shaping, but others are ones I don't feel too big on using as they are often a less interesting "Make a DC 10 Wis or Cha save every day or find your alignment changed" unless you leave or get Remove Curse or Dispel Evil or Good cast that applies to planes like Bytopia, Abyss, Baator, and Mechanus. In many cases they gloss over some residents of the Outer Planes, so really no mentions of Archons (nothing on the 4e elemental ones either), 2e Eladrin (4e ones are presented later in the book as a sample subrace) or Guardinals, though do say there are celestial Elves in Arborea. They do mention the Outer Planes having layers, but the only ones where they describe some layers are the Abyss and Baator.
They mention on the Outer Planes is that the parts one can travel to are the "border regions" and rest of the planes is spirit beyond mortal comprehension. I like that they've mentioned what I've argued many times in that the planes truly can't be mapped by it's arrangement is a highly supported way of conceptually grouping them.
For the Inner Planes they actually got a lot of coverage in this chapter. Though of what they talk about in the DMG is plenty of new material mixed in with some older stuff. They gloss over the pure Elemental planes and the Elemental Chaos from 4e which is now the "outside" of the Inner Planes. Instead most of the writeups on the Inner Planes instead are on the "border regions" with the Material Plane, which are more inhabitable than the pure elements. I think I remember Dark Sun had the elemental planes described in similar ways: where the Plane of Water is an endless sea dotted with islands, the Plane of Fire is a wastelands with cinder storms, the Plane of Air is filled has a bunch of earth motes and solid clouds and the Plane of Earth is a large mountainous area intertwined with caves. There are some mentions of para-elemental regions in these elemental border regions too. The Genies are described as living in these border regions, though only the City of Brass gets a good writeup in it all. In the landmarks there's places like the Sea of Worlds, Aaqa, Frostfell and the Isle of Dread. I suspect they're going to use a lot of this material on the border regions for their next adventure path involving Elemental Evil.
The Transitive Planes are just the Astral Plane and Ethereal Planes, now as Shadow is Shadowfell which is no longer a transitive plane, where the Astral is back to just connecting the Outer Planes and the Ethereal is back to connecting the Inner Planes and the Echo Planes. There isn't much changes from what I remember on the Astral and Ethereal other than the fact that there is by implication from the ethereal curtain tables a border-Feywild and border-Shadowfell.
Which brings up the subject of the Echo Planes, where the planes last known as Feywild and Shadowfell are. While they do mention the Positive and Negative Planes as being the "outside" of the Outer Planes, they only sort of imply that echo planes are border regions to the Energy Planes. Feywild has the optional rule for time flowing differently, and they mention Titania being in charge of the Seelie Fey of the Summer Court and the Queen of Air and Darkness being in charge of the Unseelie Fey of the Gloaming Court. Fortunately they do mention that Seelie does not necessarily mean good and Unseelie does not necessarily mean evil, and that Hags, Fomorians and many other fey creatures aren't members of either courts. Shadowfell is described as having the Demiplane(s) of Dread. It's possible that colour can exist in Shadowfell though it gets drained away and muted by the plane, rather than 3e's description which felt to me like everything's just greyscale upon entering.
For the rest of the chapter they mention the prime material worlds of all of D&D's major campaign settings, they mention the Far Realm, and there's a writeup on the Outlands which even though they don't classify it as an Outer Plane, they do say it should be the base for a campaign set on the planes. Sigil gets a writeup in the Outland's entry and the 16 gate-towns also get mentions.
Overall they kept out rules on the really-hazardous parts of the planes, just keeping optional special rules down to 1 paragraph entries. There's certainly a lot more that could be expanded on everywhere, but it provides a good start to go with ideas on how run the planes. I'd certainly go with ideas of getting planar keys to exempt planewalkers from certain planar effects (or I can simply ignore them), and the fact some planar effects use charisma saves to avoid probably explains why some of the narrators of the Planescape material are the way they are.