These positions are good for a robotic approach, which is my favourite. I would try to come with candidate moves.
- First I would try to eliminate quickly (or find miracles in) Qxh6+ and Rg7+
- Then I would go for two more obvious checks, Bd3+ and Qe4+. I would try to get as much feeling about what's going on in these lines, find their problems to have chance to find (if necessary) some hidden properties of position (like Bd3+ is almost mate if we prevent black piece to go on f5.
- Then I would go for mate threats, trying to quickly eliminate Rg6. From point 2 I should already know that Nxf7 might be a mate threat, even if I can miss this for now and come with the move later during capture searches.
- I would go for captures and light threats, trying quickly to eliminate nonsense again. Probably here would be Qxe7, Rg8 and of course Nxf7, if not in the previous point. I guess Ng4 could stay hidden from my sight forever as the move came to my mind after writing the whole post. Engines wouldn't make such a mistake :-)
Depending on what would come from this search, which I think shouldn't take more than 5 minutes, I would try to sort candidates from the one I like the most. Well, Nxf7 is a big and obvious candidate. It threatens mate and after Rxf7 it is not even a sacrifice. So I would go for it, unless I would have even better feelings about checking lines Bd3+ or Qe4+. That is possible, as I would have already solid cover of these now. Ok, calculating Nxf7 is another story, you will do similar work like here. Find opponent's candidates, being careful, as your queen is hanging and you are a piece down. When I know only Rxf7 is serious, then I would go for Bd3+ and Qe4+ as a huge candidates. If I find some problem there, I would try to come with other candidates like Qg4. You probably don't search for all weird candidates in later moves of calculation, as your head could explode by doing that. That is a reason why we sometimes miss nice combinations. Always checks, mate threats and big captures and threats first in start position and then later, the less candidates.
In my answer to this question you can see the most brutal combination I could have in my game, but wasn't found during the game either by me, or by my opponent. Was cool to hear how lucky I am, as the computer saw it immediately. You look for calm moves in the beginning, but find the 4th calm move in a row in such a firefight!