You are right to think of Father, Son and Holy Spirit [HS] as one God. This is a key truth of the Christian faith.
What the bible would add to this, is not only are they one God, but they are also three distinct persons. That is, the Father, Son and HS are all distinct from each other, but they have the same essence or nature; one God, three persons.
Christians still want to maintain that there is one God which was a key notion of God in both the Old Testament [“The Lord our God, the Lord is one” - Deut 6:4] and the New Testament [”... for us there is but one God” - 1 Cor 8:6]. But with the coming of Jesus, this forced Christians to extend their thinking on just what God was like.
In the New Testament, Jesus made it clear that not only was he the son of the Father [John 8:42 & 14:9], but he was also God, right from the beginning [John 1:1-3 & 1:14]. The HS becomes significantly more prominent in the New Testament as well. Jesus explained that after his resurrection and return to the Father, they would both send the HS to help believers know and understand what Jesus taught [John 15:26].
So early Christians had to work out how it is that God could be one God yet be more than one person. They eventually agreed that the bible asserts there is one God, but three separate persons. We see the distinctiveness of the Father, Son and HS in Matt 3:16-17 where the Father speaks from heaven, as Jesus, the Son is baptised and the HS appears as a dove. On the other hand, we see the unity of the three, when Jesus commissions his disciples to go out and tell the world all that he has taught “baptizing them in the name of the Father and of the Son and of the Holy Spirit” [Matt 28:19]. To describe this concept the church father, Tertullian, coined the phrase “Trinity” which is how most people refer to it today.
There are two traps we should avoid falling into when we think about the Trinity.
Firstly, the Father, Son and HS exist at all times together. That is, God does not change from one “mode” to another like ice, water and steam.
Secondly, we should not think of the Son and HS being secondary to the Father. While they have different roles, the Father, Son and HS are all equal in their nature.