Disclaimer: I keep getting called either a Modalist or a Tritheist on here for saying this. I will not bother answering objections. Take it or leave it.
Everything done by God is done by the Father, through the Son, in the Holy Spirit. This is the basic principle of what the Trinity means.
Obviously the Father is God. But if the Son really comes from God, and really gives us God's divinity (by linking us back to the Father), and really saves us, He must be God. And if the Holy Spirit really comes from God, and really gives us God's divinity (by linking us back to the Son), and really saves us, He must be God. Therefore the Father, the Son, and the Holy Spirit are all really God - they are of one essence.
Obviously what is done by God is done by the Father. But it is always done through the Son. The world was made through the Son, the voice of the Father is conveyed through the Son by the prophets and the angels (so much that the New Testament frames the "Adonai" and "YHWH" of the Old Testament as being Jesus specifically), and we can reach the Father by the intermediary of the Son, the sole mediator between God and man. One cannot separate the actions of the Son from that of the Father, because, generally speaking, if the Father is God acting "distantly", then the Son is God acting "among us".
And likewise, everything done by the Father through the Son is done in the Holy Spirit. It is in the Holy Spirit that the world was made, it is in the prophetic "Ruach" that the prophets could speak the words of the Father and the Son, it is by having the Holy Spirit in us that we can have the Son, and, through the Son, have the Father. One cannot separate the actions of the Holy Spirit from that of the Son, and therefore from that of the Father.
I'll note also that in the scriptures, the Son is described as the image and presence of the Father, and the Holy Spirit as the image and presence of the Son. But this is not like God manifesting Himself under a different form for the sake of His economy, but rather a single "movement" of the Father "reeling us" back to Himself, through the Son and the Holy Spirit. That is also why St John calls the Son the first Paraclete and the Holy Spirit the second Paraclete.
All this being said, my point is: the Father, Son, and Holy Spirit are of one energy. They do not have three energies, wills, loves, justices, etc. that coincide, but truly one.
Finally, the Father, Son, and Holy Spirit are three distinct and different persons. Even though they are one in essence, and one in energy, they are three persons, distinguishd by something proper to each. The Father is the sole cause, the Son is begotten of the Father alone, the Holy Spirit proceeds from the Father alone. The idea of the Word of God coming forth from God and being God manifesting Himself among the Israelites, and of the Breath of God coming forth from God and being God manifesting Himself in the Israelites, was not foreign to Judaism, but what the New Testament makes clear once and for all is that these two are not mere impersonal powers, or maybe angels carrying those powers, but they are truly persons, and truly God. Is this compehensible? Hardly so. But it can be experienced, through the sacraments. By tasting of the Eucharist, we -know- that the Father, Son, and Holy Spirit are three persons who are one in essence and in energy, even though we ultimately reach a paradox if we try to write it down. Incidentally, pretty much every Trinitarian heresy has been an attempt at making sense out of this mystery.
>Christianity often gets accused of polytheism, especially by Muslims, while I do understand that's not the case I almost feel like I can't really explain in solid terms as to why that is the case, it's three different persons of the same essence that make up God to my understanding, but many times we make sure to worship all three persons, just something as simple as crossing oneself is a sign of worship of all three persons at the same time innit?
The three divine persons don't "make up" God. God does not have parts to be "made up" of. The Father is fully God, and the Son is fully God, and the Holy Spirit is fully God.
>So how is it not polytheism then when it's worshipping three seperate and different entities in their own right who can act independently of one another?
The Father, Son, and Holy Spirit are distinct but they are not separate and do not act independantly of one another. They are not three persons that co-operate, but they truly have only one operation. Again, what is done by God is done by the Father, through the Son, in the Holy Spirit (and, by the way, you can see here why the Father is most often called "God" while the Son is called "Word of God" and the Spirit "Spirit of God").
(cont)