how do I improve my vision to look 3-5 or more moves ahead?

Most of those 6 movers on Tactics Trainer are not particularly diabolical. 

This was the last one you missed.  It was only a 4 mover. It starts with black playing Rf8. You had to play white.

Solving it is a matter of recognizing the pattern. It says you got the first move Bc4+ right. After the tactics trainer responds with Kh8, the rest should be easy if you recognize the pattern.

I second this - I completely agree. Tactics help with calculation, but a lot of tactics are intuitive or pattern-recognition-based; endgame studies get you thinking deeply about every move. I find that I calculate longer in delicate endgames then in any other part of the chess game (unless I am calculating some forced mate sequence of course).

jaiminsonu, this puzzle is one of those forced mates (pattern-recognition for this one though). Your idea of the h-file being important was spot on (why you played Rh3), but ...h6 hangs on for Black. Sacrificing the Queen with Qxh7+ FORCES checkmate by not giving up the initiative (not allowing Black time to defend). 

That's how improvement works. You work and work and work and try so hard, and fail.

Then you look at the answer and learn.

If you do this every day, then years later you will be a good player. This is how it works for everyone.

As a tip, for tactics you need to look for forcing moves. The 2nd move Rh3 is a threat, but black can defend. You have to consider forcing moves no matter how suicidal. Qxh7 is the most forcing, so you must consider it..

Yes, well, in the past my verbosity was rivaled only by the mighty Elubas whose walls of text were rumored to bring entire nations to their knees.

But I have tried, over the years, to cut my posts down.

I very very very rarely almost never calculate 5 moves ahead in the middlegame

Yeah, 5 full moves (10 ply) is rare.

Although beginners usually mean 5 ply (2.5 moves) which I calculate on practically every turn.

Back when I used to teach chess (decades ago), I would teach the tactics in "family groups".

There is a whole family of tactics and mates based on the weak back rank, another group based on the broken fianchetto or weak color complex, another group based on the advanced Pawn on f6, etc.

Here are a few of the latter group (mates based on an advanced Pawn on f6), with the easiest ones first:

@Bluemu, This puzzle is wrong. Black lose because black played Kh8??.

Why not black play Qd8!!! That should be draw.

@blueemu

Yes, and the third one is even worse. Stockfish gives a -3 evaluation after 2..Qxc3!

So, Bluemu failed 2/3 his own puzzles. (66% failed rate, lol)

See, O.P, dont get upset, it is not you alone. The other people also failed after 2- 3 moves. 😂😁😅

Bc4 Kh8 Qxh7 Kxh7 Rh3 with mate to follow.

Bc4 Bd5 Nxd5 and the threat is Nxe7 or Nf6 double check with Qxh7 mate next move, but also the knight is immune to capture due to the first line, so white will either win the queen or give checkmate.