I understand that your choice in Moscow is basically Gett, or Yandex-Uber joint. I think you need a specific app for your smartphone, stock Uber app will likely not work. Yandex has Minivan, not so sure about Gett. Mind that you need data connectivity on your smartphone to use either one.
I remember that you are traveling with children. Please note that Russian driving laws require children of ages 7 or less to always be in child seats. This applies to taxis as well. Many taxis have a child seat, and you can specify this when ordering a car via app, but they would likely not have more than one child seat, apart from very specific "kid taxi" services. If you have no more than one child of that age, or none, this is not of concern.