Place the farro and water in a medium-sized sauce pan. Bring to a boil, then turn the heat down, cover, and simmer for 20 minutes. Drain the remaining water from the pot, add the olive oil and salt, and toss to combine. Allow the farro to cool to room temperature.
Add the cherry tomatoes, mozzarella, basil, and arugula to the cooled farro. Toss everything together.
Whisk together the balsamic vinegar, red wine vinegar, olive oil, and salt and pepper.
Pour the vinaigrette over the farro and toss to coat everything evenly.