New at chess - how to get started and improve?

Indeed, this is quite the most common topic in the General Chess Discussion forum (sometimes even appears in Lichess Feedback). Often enough, answers are short and flat, as people don't want to spend 2 hours answering this in great detail. These answers often sound like "do tactics and puzzles", "get a book or watch videos", "learn the opening rules", "join a club" or whatever. Nowhere specific, so I finally decided to spend these 2 hours and describe more elaborately on what you should focus on! Btw, this is based on experience of accompanying streamers on their way from 800 to 1400 or from 1100 to 1700.

In the following I'll assume a person who just understood the rules of chess, and now is like 600-1000 rated in rapid or classical (maybe provisional still). I hope the following helps everyone new at chess, or also those who are stuck at low beginner level for a long time. Obviously, what you make from it all depends on you.

An absolute mental requirement is to accept that you need patience and you can't climb the rating ladder in just a few days or weeks. Chess is a long-time commitment especially when you're absolutely new to it. Now I'll describe a typical pattern on how to develop skill. This is not to be understood as a fixed order, it's a continuous process. What makes all of these steps successful is regular practice. Play rated games in the pool with slow time controls, preferrably with increment. Do a bunch of puzzles every day (or at least every 2nd day :-)). Don't go overboard with puzzles (quantity vs quality), play them considerately. Lichess also offers some basic training material in lichess.org/practice. Your improvement will transfer from slow time controls to faster ones, not the other way round. I'd say don't blitz too much, and definitely keep your hands off bullet as long as you're lacking experience for fast play.

First step: Stop blundering pieces

(that is, straight blunders must be less and less frequent). The very first thing for beginners is to not put pieces on unprotected squares where they can be captured directly. This is the part that will frustrate beginners the most. But your opponents are not better so often you'll also get the happy opportunity to snack a free piece of theirs.
Moving to a covered square is not the only way to straightly blunder a piece. It's also if you overlook that a piece got attacked by the opponent's recent move, or that your own move leaves some other piece unprotected.

Another first step: Make sure you know ALL of the rules. This concerns en passant; when castling is allowed and when it's not; threefold repetition, 50 moves and stalemate.

Second step: Introduce yourself with basic mating patterns

This commonly addresses mating with a queen (don't stalemate!), two rooks (or rook+queen) and just a single rook. These endgames are mostly covered by lichess.org/practice. Also, very very important: backrank and basic queen infiltrations (Ng4+Qh2, Bd6+Qh2, Bc6+Qg2, pawn f3/h3 + Qg2 etc).

https://www.youtube.com/watch?v=iBZLU1FXhcI

I think this is a pretty beginner-friendly source for important middlegame mates!

Third step: Be aware of the typical value of pieces

The difference between rook and minor piece is considered 2 pawns on average, in fact rather tending to 1.5 pawns. Two minor pieces are most definitely stronger than a rook. A queen is better than rook+minor piece, and roughly equal to two rooks (convention says +1 for the rooks. but it highly depends on how safe the king of the rook party is).

Minor pieces are roughly 3 pawns. But the less pieces remaining on the board, the more important the pawns. If there are no pawns left, you need at least a rook to win. Knights and bishops are said to be about equal, but it depends on the type of position. Bishops prefer open positions due to their long range, and knights prefer closed structures due to their unique abilities. Also knights can't hold a protection when they're attacked and expelled. Rooks need open files, they're good attackers but clumsy defenders.

More importantly for you, the value of pieces also comes together with basic protection rules. How often is a piece attacked by the opponent and how often is it protected? But also, what's the priority of pieces, and are they in each others way or not? For example, it makes no sense to:
a) protect a square with queen and king when it's attacked by two minor pieces.
b) attack a pawn with two pieces when it's protected by another pawn.
c) protect a piece with your queen and a rook/bishop standing behind when it's attacked by two lower-valued pieces. As the rook/bishop is standing behind the queen, it won't be the first to recapture and you'll definitely lose material.

Fourth step: Get familiar with basic tactics

Alongside the basic principles on what makes a piece hanging, this addresses pins, forks, skewers, discoveries, fake protection, attraction and distraction, destructive sacrifices to mate. Even at higher level, a ton of games is decided just from one of these basics. Lichess has excellent tools for both general tactical practice and focusing on specific motives. I can also only recommend the daily puzzles as part of your training routine. You will always learn something from the daily puzzle, even if these are too hard for you at first.

Fifth step: Get into basic endgame theory.

Endgames with a single pawn, remote and protected passed pawn, breakthroughs, winning with queen against 7th rank pawn (or knowing when it's drawn), rook + pawn vs rook, the wrong-squared bishop, knight vs border pawn etc.
They all complain to their coaches and master idols: "please help me in this or that opening" ... but the endgame is where wins, losses and draws are decided!

Sixth step: Healthy opening

We're not talking about specific openings yet. Just about the golden rules, to claim the center, free your pieces, get all your pieces into play, castle and connect rooks, make all your pieces as effective as possible. But also watch out for tactics immediately, especially don't get scholar-mated or lose material early. Of course, one could object that opening hardly matters at beginner level because games are still blundered and reblundered at any time. But a good base to start is still absolutely recommendable, it allows for comfortable wins and avoids early disasters.
If you're sick of randomly putting your pieces into a knot, or your queen being chased around the board, or getting crushed because you forgot castling, you must learn the golden opening rules immediately.
If you have the need to get to know common openings, feel free to, but don't emphasize on it too much.

What's next?

At this point you should have established a rapid rating of I guess at least 1500 and you should be happy with your play even when you have phases of stagnation. At this point it's up to you how much you're determined to keep improving and get into the advanced stages of positional play, weaknesses, imbalances, offensive and defensive patterns, initiative, endgame activity, an opening repertoire etc. Books, videos, streamers, OTB clubs are good sources for that. Maybe you'll develop dedication to play arenas regularly, get engaged with a club or streamer community, try chess variants for more fun. But first, I absolutely predict, you will need several months to complete the basic steps. Happy chessing!