The main independent social media platforms I’m familiar with are Mastodon and Bluesky (alternatives for Twitter), and Pixelfed (for Instagram). They all avoid the hazard of unaccountable billionaires by allowing for multiple servers to form a federated network. They also eschew advertising and engagement-driven algorithms to create a less hostile experience. There are also platforms such as PeerTube (for YouTube), WriteFreely (for Medium), and Friendica (for Facebook) but I haven’t used them. (These other ones all use the same protocol as Mastodon and Pixelfed, and form what’s called the “Fediverse”.)
I’m most partial to Mastodon (@alec@perkins.pub) but am also on Bluesky(@alec.perkins.pub). Now that I’m finally quitting Instagram I might be on Pixelfed soon. Also, more direct contact methods are available here: alecperkins.me.
Note that most of these independent platforms have ways to import your data from Twitter/Instagram/etc, or at least your network of connections, so before you delete your account there make sure to download it or check out how to import it first.
Bluesky is very close in experience to what Twitter used to be, with additional moderation and curation features. The community and vibe are also similar (pre-Xification), and it's good for keeping up with news and culture in real time. Signup is easy since it’s one server. Once you sign up, there are starter packs to quickly find a variety of other people to follow. There are also different feeds made by other users which you can subscribe to. Bluesky still has “algorithms” in this way, but they are independently made and transparent about how they work, and it also has a plain chronological feed.
The catch with Bluesky is it’s still controlled by one company more or less, that hasn’t yet established a way to sustain itself financially. They have implemented the features needed for federation but there haven't really been others set up yet. There is some interoperability with Mastodon and the broader Fediverse, but it’s awkward. The upside to the single company is they move quickly and are very responsive to giving users ways to guard against abuse as the platform has scaled.
Mastodon is a little “slower” than Bluesky in the sense that current events aren’t as prominent. But it has been around a lot longer and has a huge variety of servers/communities to pick from, which all interoperate so you can follow across them or migrate your account. There are also more apps to use it with, like Ivory if you were a Tweetbot fan. You can get a good variety of styles of post, and there are lots of sub-communities already, like gardening or ham radio. It’s also much more international, places like the EU actually run their own server for goverment accounts. Mastodon is very hashtag centric, and you can follow different hashtags to keep up with particular topics. I’m fond of #LichenSubscribe.
The catch with Mastodon is you have to pick a server to start, and there are many to choose from. This is a strength and a weakness: it allows users to find a server that fits their values and moderates content in the way they like, and gives them flexibility if the server isn't working how they want. But, it does mean the servers can block each other and users can get caught up in whatever the latest intercine dispute is between server admins. Servers also have to individually figure out how to financially sustain themselves, with varying degrees of success. The Mastodon team has done a decent job of picking a starting server easy, and you can always move later. However, this is still a sticking point. I ended up running my own server because I can and it means I have complete control and never have to worry about who is governing my experience. This requires some know-how though. (I have been considering setting up a server for Hudson County organizations and people, but haven’t established there’s a demand for it yet.)
Mastodon also has issues with an "HOA crowd" as it’s known who feel a need to police how other people use the platform. Many server moderators do not tolerate that behavior but it still shows up from time to time. For example, Black users have been harassed for not using content warnings when talking about their lived experiences with racism.
Pixelfed is a lot like Mastodon, built on the same protocol so users can actually follow each other from across platforms. Its interface is closer to Instagram though, oriented around sharing photos. You also have to pick a server to start with but they have a helpful list. It hasn’t been around as long and has a small team (one person?) so it’s taking longer to establish. There is an iOS app coming soon though! And it supports importing your Instagram content.