I would say that obtained is not the word you're looking for. It's so vague, so three-day-old-beer flat, so general, that it says little.
Rather, I'd say:
I bought a new car.
I got a new car for my birthday.
I won a new car in a lottery.
Use a vivid verb instead of something olive drab and camouflaged. However, I obtained a new car is grammatical, understandable, and boring. But it does invite questions: "How?", "Where ?", "When?", "From whom?"
The second case is just plain wrong. You can acquire a skill or a language, for example, but you can't obtain one. Those things aren't like advanced degrees -- it used to be that you had to earn a PhD, an Ma/MS, BA/BS, but now you can obtain them from dealers who print them and sell them to anyone willing to buy one.
I learned how to play the piano.
Saying "I obtained piano playing skill." is like talking with wads of cotton stuffed in your nostrils. It can be done, but it makes the folks you're talking to feel strange and makes you sound even stranger.