What is "на" and why is it needed? I saw another sentence like this but with "у'' instead of "на." What do they mean?
на (on) + locative with musical instruments, у (in) + accusative with sports
Грати на піаніно, грати у гольф (to play golf)
Does піаніно remain unchanged in locative?
It remains unchanged in any case. There are a lot of such neuter nouns that end in o and are undeclinable: авто, метро, доміно. All of them are loanwords
Why is it "піаніно"? Is it some kind of diminutive of піанo?
We don't have an instrument called піано :)
But it's from from italian pianino (small piano). This must be the big one http://goo.gl/jy736h, so this is small http://goo.gl/MgqvzO. That big instrument (grand piano) is called рояль, small one піаніно and the whole class is фортепіано
How far the sentence Я хочу грати в теніс. is correct?
Shouldn't there be a definite article for "piano"?
I got Tenis wrong in a multiple choice question. Does it need another preposition or something?
There's more than one correct translation from Ukrainian to English I wrote I want to play on the piano which is also correct but it marked me wrong