Vox Machina: More mature and 'Korra-like,' contains explicit callouts to Avatar (including a Janet Varney cameo), and is just a darkly funny and wholesome D&D adventure show. Scratches that same fantasy itch.
DOTA: Dragon's Blood: Animated by Studio Mir in the same style as Avatar, another fantasy epic, and mature and fast-paced. The plot is kinda wild, but it still feels 'western' for lack of a better word, and I really like the characters and their VAs. There are some explicit callouts to Avatar too (like Luna's characters design and magic that looks a whole lot like bending).
Arcane. Not a direct inspiration, but it's simply really good and "magitech"
Pantheon. Not actually very similar to Avatar, kinda sci-fi, but watch it if you haven't. I think this is the one 'cartoon' I rate significantly above Avatar.
And some more 'mixed' recommendations I bounced off of but many fans swear by:
Dragon Prince: The most ATLA-like, produced by an ATLA writer. As shallow as it sounds, I bounced off because of the animation style, and it felt kinda immature, but its long running and very popular among ATLA fans.
Voltron: Legendary Defender: Sci fi, but animated by Studio Mir, and has a similar "epic fantasy" feel. I actually liked this one, but just forgot to pick it back up...
Classic cartoons like gravity falls, she-ra, owl house, steven universe, adventure time. If you're really into it, LoK paved the way for LGBTQ relationships in many of them. I also enjoyed Young Justice and Justice League, while I'm throwing those out.
There are a number of graphic novels that continue the story, so you can kinda watch the gaang grow up. The first series, I believe, is called The Search, and it's about finding Zuko's mom. There are also a ton of spinoff novels about the other avatars if you want more lore.
Search is the second one. Pretty good. The first set of graphic novels is the Promise if I remember correctly. It picks up right after the kiss. Not to get political, but there are aspects of it that resemble the Israel and Palestinian conflict and may give you some thought provoking ideas. I'm just clarifying because I didnt want you going into the comics reading the first dozens of pages and realizing that Zuko's mom's story isn't there.
There isn't an animated continuation of Aang's story. The only other animated series in the Avatar universe is Korra, the avatar that follows Aang.
Korra takes the story through some very controversial changes, and production was frought with difficulties thanks to Nickelodeon constantly changing their minds on what they wanted. Season two is widely seen as the worst in the series by some distance, but season 3 redeemed it somewhat.
It's worth a watch for sure, just don't expect it to live up to the magic of the original.
Korra is great, it had some issues from Nickeoldean that reduced its success. Basically they had no idea if there was going to be a season 2 or 3, so it was written that way.
3 and 4 are more connected.
There are 3 books series, each about Kyoshi, Yangchen and Roku (released in that order). Good books.
There are a lot of graphic novels for both Aang and Korra. I know they are free on webtoon and Hoopla (many libraries offer free acess to hoopla). (Webtoon doesn't have all of them yet, they've been regularly posting in the past year)
Netflix ATLA is a live acation remake made by different people. General option is meh, not as bad as the movie.
The movie is probably the worst movie released by a major studio, even by today's Netflix series wrecking standards.
I don't get the controversy around Korra. I thought it was brilliant. I assume it's because people wanted more of the Aang Gaang and it was something different (and maybe a little bit because it was a girl this time).
I really liked Legend of Korra, personally. Not as much as ATLA, mind you.
Like others said, the story isn't really a continuation of ATLA, it just occurs in the same universe. You do see a little bit of how the characters from ATLA few up and their own offspring, but it's not really the focus.
It reminds me a little of Kevin Smith movies. They all happen in the same world, but they're all their own thing.
One thing I didn't like about Korra was that each season seemed kind of self-contained. ATLA had a better defined arc between the seasons in the whole series. I think they did that in case they got cancelled and they didn't want to end on a cliffhanger, so I can appreciate that.
I liked exploring the focus of a more mature avatar growing into her role.
I've never watched the live action series, though. I'm a little hesitant after seeing the movie...
Personally, I would watch Korra first then get into the live action series, simply due to the fact that watching the same story twice in a row seems like a bit of a drag.