Is this a correct sentence your grandmother has always given pianno lessons?

Yes (although "piano" has only one N).

The present perfect aspect in English means that the action began at some point in the past and continues - she used to and still does give piano lessons.

There are some who will tell you that "always" cannot split the verb phrase as you have it doing, but that is nonsense. Single-Word adverbs have always fit very nicely after the first auxiliary verb in English.