We've been to the Vatican on our Rome trips — the basilica, the square, the museums on the slow day. We don't have a dedicated trip to Vatican City as its own destination because, practically, it's been an afternoon-walk-from-Rome thing both times. Same advice as everyone: get there early, book the museum skip-the-line, and don't try to do it in the same day as the rest of Rome.
Vatican City
Derek Johnson's trip notes from Vatican City: St. Peter's Basilica, the Sistine Chapel, and the smallest country in the world inside Rome.