Odwiedzać takes accusative, so this is the accusative for 'nasza rodzina'.
"Nasza" is the nominative, and "rodzine" is not a word. You cannot forget about the 'tails' in ą and ę, often you can either change the word (kąt - corner, angle; kat - executioner) or at least come as a person who does not know grammar, as often the only difference between the correct and incorrect form of the word may lie in the presence/absence of the tail.
Not exactly. It's a correct sentence, it makes sense, but it's not "I have visited my family".
So far, you have only learned imperfective verbs. Such verbs are general, habitual. The action which they describe is either not-finished, or we don't know if it's finished (or sometimes it means that it doesn't matter). So "odwiedzać" is imperfective. The perfective verb, which means that you 'succesfully' visited someone one time, is "odwiedzić".
Therefore: "Odwiedzaliśmy naszą rodzinę" is "We were/have been/had been visiting our family". Many times.
And "We have visited our family" (one time) is "Odwiedziliśmy naszą rodzinę".
And you wrote "I have visited", which is "Odwiedziłem" (if the person speaking is a man) or "Odwiedziłam", if the person speaking is a woman. Yes, the gender is visible in past tense verbs.
Actually it could be also visible in plural: "odwiedzałyśmy" (were visiting) and "odwiedziłyśmy" (have visited), if "we" are only women.
Would it be a different word if the sentence was "We are visiting our parents?" I'm confused on the difference between the two words in Polish.