Doing puzzles at lichess.org I stuck at raiting 1750 +- 50. I cant really go over 1800. So what are the first things one should check by solving puzzles? Are there some tricks how to solve puzzles?
I had hit a plateau like you. Probably spent about two years at the same tactical rating level until I watched a video on chess.com (it's a series, actually, by IM dpruess "Exercises to Become A Tactical Genius") that reshaped my thinking entirely.
What I used to do was analyze each of my pieces, one by one, and explore the possibilities of each piece. It was almost a guess-and-check type methodology. Often I'd get the puzzle correct, but I took so long at doing it that my tactical rating would only go up only one point. In a game, doing this, I'd surely lose on time before arriving at the correct combination.
The video I'm referring to got me thinking in terms of concepts first and only then explore those moves that would play to the concept that stuck out the most in the position.
My recommendation to you is to first familiarize yourself with a complete list of tactical concepts. Then, when you are looking at your next tactical position, see if you can spot the concept or concepts that might be featured in the position so that you can look for ways to exploit that feature. For instance, if you notice a piece is pinned, see if any moves jump out at you that will make good use of that fact. In my experience, this technique will save you a lot of time because you'll find yourself only analyzing a small subset of possible moves and the odds that the right moves are in that subset will be high.
As a word of caution, be prepared to see your tactical rating take a big dip while you get used to this new way of thinking, but I believe that once you get the hang of this way of thinking, you'll see your tactical rating shoot up. For instance, I found my rating plateau at 1900 for a couple of years, changed to this way of thinking and saw it drop to maybe 1800 and then shoot up to 2300. All the best to you. I hope this helps.
The correct way is to evaluate the positional strengths and weaknesses and find candidate moves based upon this evaluation. Another player complained that by knowing it's a tactical puzzle, you should just check the most outrageous move first. I complain that no one whispers in your ear that a tactic has appeared on your board.
I find it helpful to start by searching for forcing moves (where only one or two moves are legal, in case of a check, or nearly required, in the case of losing a queen), if for no other reason than it lowers the set of possibilities you consider before having to move on to a broader search.
Thinking through the process of creating a puzzle, a major challenge would be ensuring that there weren't better moves, or a different solution. Where the puzzle isn't a recreation of a historical position, the creator likely worked backwards from the final step and constructed the board around the path to that step. Because of this, many puzzles end up being a train of forced moves, where you're looking for fork/skewer/pin tactics to get to the final solution.
This could feel too much like studying for the quiz, rather than improving your actual chess skills, but I would argue that starting your analysis of the board in an actual game should also start by analyzing any available forcing moves.