Do Christians ever pray to God the father or the Holy Spirit?

I find the idea of the Trinity to be very confusing. It seems to me that people pray to Jesus but never to God the father or the Holy Spirit. Why is that? If someone prays to God, does that mean the request can be fielded by any combination of the three?

One thing that occurred to me is that Jesus may be in charge of PR, with the other two working in the background. Jesus would be uniquely qualified for this position, having been part human and spending time on Earth.

Another possibility is that God the father retired, with Jesus taking over the family business. The Holy Spirit might be in charge of handling the power transition.

Does the Holy Spirit have a gender? I always see it being referred to as “it,” uncapitalized. If I were the Holy Spirit, I would register a complaint.

Before Jesus, was there a Binity between God the father and the Holy Spirit? Did they cooperate in creating the Universe, or was that just the work of God the father?