Look for forcing moves first (Checks, captures and attacks).
It's hard to explain in text form. Most likely you could find instructive videolessons on Youtube.
I remember, John Bartholomew has several videolessons where he's solving puzzles on chesstempo. He's explaining his thought process well.
I was making good progress on the puzzles, but recently I seem to have regressed. I am not sure why, but when I fail I get some benefit from looking at the computer analysis and learning how my selected move is inferior to the solution.
I am wondering whether you have any general advice on how to reason about chess puzzles. What sort of thought processes should you go through before deciding on a move?