Bxg6 wins immidiatly, cause its quite easy too see that Black cant take back without getting mated. But if he doesnt take back, his king also lacks on squares and gets mated fast.
Methodology to figure out order of moves?

I let the engine play the game and a mate is not within 10 moves yet alone 5 moves. Sorry, I don't see that far.

@Epilogue, there is a slight problem. The initial position you show is not the actual position to be evaluated. If you actually go through the problem, that position is BLACK to move and it starts out with Qf5 (I was wondering why @mostruash had Qxf5 in their solution, and this preliminary ... Qf5 move is why.
@mostruash, as to the solution given, while there is no forced mate at the end of the sequence, white has won black's queen. This is a significant loss of material and, even though white is now only up a queen for a rook, this should be enough to win the game.
Not all puzzles lead to a mate. Some puzzles need you to find a gain of material. Some even give you a seemingly losing position and need you to prove that it is actually a draw through repetition or stalemate. It takes a lot of practice to be able to evaluate a position to determine what the puzzle actually wants you to do. You'll get better with time. As others have said, however, often a good start to solving puzzles is to look at the most forcing moves first - checks, captures, and threats. Even if those don't immediately lead to something, it might give you an idea - for example, maybe an immediate capture doesn't work, but if you make another move first as a setup, then the capture will work the next move.

@Epilogue, there is a slight problem. The initial position you show is not the actual position to be evaluated. If you actually go through the problem, that position is BLACK to move and it starts out with Qf5 (I was wondering why @mostruash had Qxf5 in their solution, and this preliminary ... Qf5 move is why.
@mostruash, as to the solution given, while there is no forced mate at the end of the sequence, white has won black's queen. This is a significant loss of material and, even though white is now only up a queen for a rook, this should be enough to win the game.
Not all puzzles lead to a mate. Some puzzles need you to find a gain of material. Some even give you a seemingly losing position and need you to prove that it is actually a draw through repetition or stalemate. It takes a lot of practice to be able to evaluate a position to determine what the puzzle actually wants you to do. You'll get better with time. As others have said, however, often a good start to solving puzzles is to look at the most forcing moves first - checks, captures, and threats. Even if those don't immediately lead to something, it might give you an idea - for example, maybe an immediate capture doesn't work, but if you make another move first as a setup, then the capture will work the next move.
ok
Ok lol

See this puzzle: https://www.chess.com/puzzles/problem/132016
How do I figure out the best move for me and also for my opponent so that I can calculate 3-4 moves ahead? There are way too many possibilities without filtering out some.
I get quite overwhelmed in such positions.
You're rated 651. You're not seeing 1 move ahead, let alone 3-4 moves ahead. When it comes to tactics, you always want to look for Forcing Moves.

Couple thousands of puzzles later, I'll answer my old self.
- 3 pieces versus 3 pieces, but black seems to have an edge due to extra defender that is the King. But a removal of a defender can allow us to have a material win.
- Black knight is defending two pieces. So if you attack one of them with a forcing move, the other one will be left undefended. Luckily in this setup, black queen is not defended by the black king, which allows our attack to succeed.
- How to see the solution to this puzzle? When multiple pieces threaten each other, the solution is usually related to removing a defender, or overloading a defender.
See this puzzle: https://www.chess.com/puzzles/problem/132016
How do I figure out the best move for me and also for my opponent so that I can calculate 3-4 moves ahead? There are way too many possibilities without filtering out some.
I get quite overwhelmed in such positions.