I want to preface this by saying that I've only recently gotten into D&D and I'm not familiar with a lot of the rules and mechanics, but I'm trying. I've been building a universe for a while that I wanted to adapt into a book or some other medium, but since getting into D&D, I thought that making it suitable for campaigns would be a good idea.

My initial planning has multiple planetary systems in a single galaxy, and potentially multiple habitable planets/worlds in a system. While the vast majority of everything would take place in a single world, I hadn't really considered the other planes of existence... Would each world/planet have it's own independent planes or would they each be interconnected? For example, given our planetary system of Sol, would Earth and Mars share an Ethereal Plane with the possibility of travelling between the two planets that way, or would they have their own distinct planes?

Is this something that's covered by any official material or is it up to the DM's discretion?

I think in theory the Inner and Outer Planes are the same for all the worlds on the Material Plane, except a few weird ones like Eberron. The planes are all infinite, which could account for differences between worlds. But I’m not 100% sure about this.

Ask your dm.

In general, The forgotten realms and a handful of others (Ravenloft and Mystara primarily) are connected as described in the dmg and in the ravenloft sourcebook.

Most others have their own cosmologies, Ebberon(With Kyber, Sibrys, and the 13 moons), Theros(With the 6 wards of the underworld, and nyx), and Ravnica are good examples. They are only connected if your dm wants to build a plot around it.

At least the way I run them, the Outer Planes are the same for pretty much every world. There are exceptions, but they're rare. The Inner Planes, however? Each world has its own version. This is mostly just common sense, considering that the Shadowfell, the Feywild, and the Ethereal Plane are based on the geography of the Material Plane.

Quote from Thauraeln_The_Bold >>

At least the way I run them, the Outer Planes are the same for pretty much every world. There are exceptions, but they're rare. The Inner Planes, however? Each world has its own version. This is mostly just common sense, considering that the Shadowfell, the Feywild, and the Ethereal Plane are based on the geography of the Material Plane.

 Hmm. I like that idea.

Quote from PenelopeTheWeaver >>

Quote from Thauraeln_The_Bold >>

At least the way I run them, the Outer Planes are the same for pretty much every world. There are exceptions, but they're rare. The Inner Planes, however? Each world has its own version. This is mostly just common sense, considering that the Shadowfell, the Feywild, and the Ethereal Plane are based on the geography of the Material Plane.

 Hmm. I like that idea.

 Or you could just say that the Inner Planes (including the Feywild and Shadowfell) are infinite and the different ways they are perceived on various worlds are the result of regional differences within the planes.

You might consider reading up on some Spelljammer and Planescape lore, as those are two D&D "settings" that share the main idea of traveling between other settings (meaning different "prime material" planes). And by "might consider" I mean "really should" because it covers the exact subject you're inquiring about. Just do a google search for the titles, there's a fair bit freely available on both out there.

Quote from TheNorsePantheon >>

I want to preface this by saying that I've only recently gotten into D&D and I'm not familiar with a lot of the rules and mechanics, but I'm trying. I've been building a universe for a while that I wanted to adapt into a book or some other medium, but since getting into D&D, I thought that making it suitable for campaigns would be a good idea.

My initial planning has multiple planetary systems in a single galaxy, and potentially multiple habitable planets/worlds in a system. While the vast majority of everything would take place in a single world, I hadn't really considered the other planes of existence... Would each world/planet have it's own independent planes or would they each be interconnected? For example, given our planetary system of Sol, would Earth and Mars share an Ethereal Plane with the possibility of travelling between the two planets that way, or would they have their own distinct planes?

Is this something that's covered by any official material or is it up to the DM's discretion?

So the PHB and DMG give a sort of "default cosmology" which some folks treat as canonical lore and others treat as simply a suggestion to draw from for their own world building.  One thing I'd suggest, unless you see your campaign jumping into your game cosmos's equivalents of Earth and Mars, I'd keep that stuff up in the air.  DMG give excellent advice on scaling world building by player level and having things like what's up beyond the sky is one of those things PCs likely wouldn't be equipped for realistically till tier 3 without significant NPC or patron handholding.

That said it sounds like you're prime material plane largely works like the IRL universe where planets are attached to star systems as part of a galaxy.  Spelljammer posits that material plane world are contained within "crystal spheres" which float amidst a cosmos of phlogiston and is something many gods can't or won't touch.  Depending on how active outer planar powers are in your prime material worlds, it's possible all or many prime material worlds adhere to the same cosmological understanding of the multiverse with identical terminology likely imposed on them by planar powers.  It's also possible that different worlds intersect with the same multiplanar cosmology but have differing experiences or terminology for it (notice fiends, and maybe celestials don't make their true names public. they could have as many aliases as there are worlds.  Or you could have multiple outer planars that don't interact much with each other but particular planar configurations intersect particular worlds.  

Other planes of existence in 5e just don't have the extensive support and definitions of prior editions (and I don't see that changing) so it's largely DMs call.  There's a recent popular work on DMsGuild (even available in hardcover) Codex of Infinite Planes.  It basically distills a lot of prior edition lore on the planes and provides some broad guidelines and inspiration to  work out your own multiverse including guidance on creating your own planes, including ways planes can connect with each other and the prime material world.  If you really wanted other planes of existence to be a big thing in your game, I'd start with that and pillage reprints of Planescape (2e) available on DMsGuild too (though Planescape is written in a world specific vernacular that takes a reader new to the world a bit to parse, think the argot used in the Mad Max movies from Road Warrior to Fury Road and you get the chant or the dark of it, as they say in those publications ... also demons and devils are called something different for reasons but it's clear what they are).