When did Tony Stark learn to fight?
Some time between the end of Iron Man and the beginning of Iron Man 2, Tony began learning to fight.
In Iron Man 2, Tony is shown sneaking in an MMA move during a boxing session with Happy.
HAPPY: What the hell was that?
TONY: It's called mixed martial arts. It's been around for three weeks.
HAPPY: It's called dirty boxing. There's nothing new about it.
Notice that Tony mentioned he'd been learning MMA for only three weeks at that point. Given this statement, and that only six months stands between the events of I.M. and I.M. 2, Tony most likely had begun learning to fight just a few months prior to the events of Iron Man 2, not too long after Iron Man.
From whom has he learned it? Is it from Steve Rogers or from somebody else?
Tony is never shown learning any combat skills/techniques from anyone, nor has it been hinted at at any point in the films so far.
In fact, as far as the boxing/sparring scene goes in Iron Man 2, there's really no indication that Happy was teaching Tony anything. Instead, it perhaps seems that Tony was taking lessons elsewhere, and then trying his newly learned moves out on Happy.
With Tony Stark being as wealthy as he is, it wouldn't surprise me the least if he had just hired a professional to give him lessons at his own convenience.
His fighting style is somewhat similar to mixed martial arts, but I'm not so sure about that.
- Iron Man 2 - Tony mentions learning MMA.
- Iron Man 3 - Tony is shown with a Wing Chun wooden dummy, suggesting that he's continued to broaden his combat knowledge/experience/training since Iron Man 2.
Captain America: Civil War - During his fight with Captain America, Tony instructs his suit to analyze Captain America's moves, and to assist in fighting.
So, perhaps Tony's fighting style is also a blend of the fighting algorithm/software belonging to whichever suit he's wearing at the time.