I think you'll get different answers on this one. Pentecostals will certainly say there are two "uses" of tongues, and that when Paul asks, "do all speak in tongues?" implying they don't, he is referring only to the "gift" of tongues listed in 1 Cor. 12. Said gifts in the list are for the church, meaning the assembled body. Thus, the reason here for all the instruction on the use of tongues "in the church." We know this portion of the letter was included because the young believers, who were gifted, were using these gifts in a way that brought confusion.
Pentecostals would also say there is the private use of a tongue, which Paul alludes to himself. This is what they would call "praying in the spirit." Here is where it gets controversial. Pentecostals say that while every Christian is not given the "gift" of tongues for the assembled body (as not all have all of the other listed gifts), they maintain any Christian can receive the gift of tongues (small g) as a prayer language. This is hard to ferret out of these chapters, but it is largely based on the scriptures in Acts where it says multiple times that "they all spoke with other tongues." It is not clear if this is so. It's also not clear if it's a different kind of tongue, or the same gift used differently in different settings. Non-Pentecostals would almost always say not every Christian can speak in tongues, and that there is only one kind.
As for whether "speaking or praying in tongues" is "praying in the Spirit," Paul seems to say "pray and sing with my spirit." Not sure about the Greek behind this though. So your question is probably going to be answered differently by those of varying traditions and/or experience. Personally, I think praying in the Spirit could include both ideas: being moved to pray the mind of God in your own language, and also praying in tongues privately. He does conclude by saying to desire that you may prophesy and do not forbid speaking in tongues. I think one could infer after these three chapters he is mainly saying, do things decently and in order.