If learning to drive is a short-term event that spans a few weeks or months, there's nothing wrong with saying:
My father learned to drive when he was 16.
If you want to emphasize, however, that learning to drive is more than a one-time event, that it's a never-ending accumulation of experiences and ongoing lessons on the road, then you would say:
My father has been learning to drive since he was 16.
If you simply want to emphasize how long he's been driving, then use the simpler:
My father has been driving since he was 16.
All three sentences reveal that your father first got behind a steering wheel at the age of 16, but they focus on three different aspects of driving: learning the fundamentals of driving, becoming an expert at driving, and just plain driving.
Now, about these two:
It is 5 years since I last saw her.
It has been 5 years since I last saw her.
Don't fall into the trap of thinking that one of these must be correct, and therefore the other must be wrong. I see that so much on the pages of ELL!
Sometimes one alternative is correct while another is wrong, but oftentimes both answers are correct, and it's just a matter of context: Are you speaking, or writing? Are you in New York, or in Sydney? Is the environment formal, or informal? I say lotsa things among friends that I probably wouldn't write in a research paper, and I've inserted many phraseologies into research papers that I would be highly unlikely to utter around friends.
I don't find any grammatical gaffe in either of the "5 years" sentences you wrote here (other than, in writing, we would usually write the word five instead of the numeral 5; we do that for single-digit numbers). However, your teacher's wording sounds too formal and stilted for casual conversation – at least in my opinion and according to where I live – so I'd probably default to your wording about 90% of the time.
As for getting some of your English lessons by watching movies, that's a two-edged sword. I'd be careful about that. On one hand, movies can give you a good feel for how people speak English in everyday life. On the other hand, not everything you read in a movie script is worth emulating. Movie directors want actors to say things in accordance with the characters in their films. So, if you watch too many mafia movies, you might end up speaking like a mobster. I don't think my wife would appreciate me saying, "Yo, Annabelle" – no matter how much she may have liked the original Rocky movie.