I would suggest looking for forcing moves first. i.e. "If I play this, he/she MUST play this. Then, I can play this, and he/she MUST respond with this. Finally, they're in a position where I can deliver mate." What I usually do look for checks, pins, forks, deflections, skewers, and sacrifices of any kind before I look for anything else. As I'm not a master yet (I'm only 14, and not close to the likes of other childhood chess prodigies), I can't speak for them, but it would help to see a 2000+ rated player's input on puzzles.
How should I solve mate-in-n puzzle?
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Sometimes you see the idea right away, sometimes even the final position, and then you just calculate to find the correct sequence.
Other puzzles are more difficult and it's useful to calculate using forcing moves like whichoneistheking said.
But even when confused you can do some general reasoning. Lets say the enemy king is on g8. Look at each adjacent square. How many of your pieces are candidates for attacking h7, h6, g6, f7, etc. This is like Vukovic's idea of the focal point of an attack. This usually helps to get a vague idea of what the final position or pattern will be like.
Depending on your ability, when you're struggling to solve the puzzle, it can also help to look at each of your non-pawns one by one. Follow their lines of movement all the way to the edge of the board (through pawns, pieces, doesn't matter). Then you may suddenly notice that a far away rook or bishop plays a key role.
I've been wondering this since I started chess.
Should I try to figure out the exact final positon before I make the first actual move?
I think I should. Then should I think like this:"If I move this piece here, his king can go here or there; if he moves here, then I play this,...then no matter what he does, I play this or that, that's checkmate."
Then, how do an expert or master think differently? Does he easily elimiate the unlikely moves thus reduce the search tree, in addtion to faster thinking process, so he can do that much faster?