Puzzle Tips
A collection of general puzzle solving tips I've come up with over the years (aimed at 1800+ solvers). I'll be adding to this post as and when I remember/come up with more 
1. Name patterns
Name as many patterns as you can! For instance, if you're attacking the king with your queen, you should be able to very quickly identify potential back rank/dovetail/swallowtail/epaulette/balestra/etc. checkmates, and being able to name them in your internal/external monologue makes them easier to recognise.
This includes words like 'pin', 'fork' and 'skewer', as well as "larger" patterns like 'Greek gift', 'trebuchet' and 'fishing pole'.
If you recognise a pattern but don't have a name for it, give it one! For example, there's a common mating pattern involving Rxg2+ Kh1 Rg1++ Kxg1: since naming it ('tube mate'), I've been recognising it much faster (another example). I've had similar success naming the 'advanced killbox mate' (another example).
2. Think about candidate solutions, as well as candidate moves
Identifying what sort of puzzle you're getting into reduces the amount of stuff you have to think about; e.g. if you're down a LOT of material you either have to mate, or draw, or pull off some insane miracle material gain -- there are no other options.
Moreover, every puzzle has features like this that drastically limit what you can do! For instance, in the opening, you should be looking to create basic patterns like pins, forks and skewers, as these are by far the most common tactics in such positions.
Similarly, in lone rook endgames there are only a few key ideas to look for: mating nets, skewers, force-trading rooks for a winning pawn endgame, or defending passed pawns to promotion.
Here's a (very high rated) example in a double rook endgame: without thinking about any particular moves at first, what could the solution possibly be? A mating net? A trade? A skewer? A promotion? Something else? Answering this question is key, and once you have it's just a matter of not allowing the opponent to stop you. Take your time -- this is a very doable and satisfying position to solve!
It's helpful to use the custom puzzles feature if you're weak on a theme that happens to be tagged there.
3. Is that piece free?
Oh nice, you win a queen in that line!! But first, take a second to see if you can't checkmate instead... and once you've done that, ensure you're not getting mated/perpetual yourself, or are already down 9+ points of material!
The same logic applies to more subtle situations.
4. Are you okay?
Since so many puzzles are about attacking and winning material, it's easy to overlook an opponent's threat on the other side of the board. Many times I've confidently made an incorrect capture before realising that the material can be won back (or worse!) with a retaliatory tactic I'd completely overlooked.
If the opponent has threats of their own then your options will be severely limited -- noticing this as soon as possible makes the solve much quicker, but again it's easy to get blindsided by that purely attacking mindset. One way of addressing this is, before even looking for candidate moves, to ask yourself what the opponent is 'saying' with their last move: "You take my piece and I'll take yours!" "How are you defending this?" etc, and how to refute it.
5. Double check is strong
Always worth calculating if you spot one!
6. Don't forget the basics
Bishops can move backwards. Some squares are not safe. Pieces don't attack the same squares after they move. Two rooks and a minor piece are worth more than a queen. Passed pawns belong behind kings and in front of rooks. Mate-in-1 is the best move.
