Yes, Jesus, the Holy Spirit and the Father, are all parts of the one and only true God there is, and who created the world and all things! We humans each have 3 separate parts to ourselves too: body, soul and spirit.
1 Thess 5:23
23 And the very God of peace sanctify you wholly; and I pray God your whole spirit and soul and body be preserved blameless unto the coming of our Lord Jesus Christ.
KJV
When we die, our spirits become separated from our bodies. Our fleshly bodies die (Jesus' body died too). Our bodies decay in the ground. But spirits do not die. Our spirits (if we've sincerely accepted Jesus as Savior and Lord (God) go immediately up to heaven upon dying.
Phil 1:23-24
23 For I am in a strait betwixt two, having a desire to depart, and to be with Christ; which is far better:
24 Nevertheless to abide in the flesh is more needful for you.
KJV
Paul here - was saying he knew that it would be more pleasant for Him to die and go to heaven - but that even so, he was willing to remain here on the earth, so that he could finish his work for God on the earth - which was that of being an apostle.
God's Spirit is here in the world, especially as a result of the great outpouring of God's Spirit, that took place on the day of Pentecost, that followed Jesus' ascension back to heaven (it took place about 1 month after HIs ascension). His Spirit communicates with people (trying to persuade them to accept Him as Lord and Savior), and much more so in the lives of those who serve God. God's Spirit is how God communicates with mankind. We don't see HIs spirit, but it exists. Similarly, we humans each have a spirit. We can communicate here on earth through our spirit as long as we are alive here on earth. It isn't often expressed in that way, but it is true that when absent from each other - we communicate through our spirits (in other words - since we aren't dead, there are still ways to communicate with others while absent from them, for example: (1) by letters, (2) by phone, or (3) by communicating through the internet, such as through social media like this forum.
Col 2:5
5 For though I be absent in the flesh, yet am I with you in the spirit, joying and beholding your order, and the stedfastness of your faith in Christ.
KJV
2 Cor 10:10-11
10 For his letters, say they, are weighty and powerful; but his bodily presence is weak, and his speech contemptible.
11 Let such an one think this, that, such as we are in word by letters when we are absent, such will we be also in deed when we are present.
KJV
Col 2:6
5 For though I be absent in the flesh, yet am I with you in the spirit, joying and beholding your order, and the stedfastness of your faith in Christ.
KJV
I believe God's "Father part" - represents His decision-making part. Jesus was always careful to pray and rely on the Father's guidance in HIs life. We who serve God are also to do so - we are enabled to do so because of Jesus' death on the cross, which figuratively has"removed the middle partition" (partition - the curtain their used to be in the temple under the Old Covenant) - so that we can come into the Father's presence, without fear, due to Jesus now being our mediator (the go-between between God and man).
Heb 4:16
16 Let us therefore come boldly unto the throne of grace, that we may obtain mercy, and find grace to help in time of need.
KJV
We can address our prayers to God, by any of HIs names (Father, Jesus, Spirit) - I'm sure it's ok. I know of accounts of God's people's communication as using all of these terms in the Bible. All the terms refer to HIm as being one God, and one being. It's true that Jesus addressed His prayers to just "The Father" - but I think this was to be an example to us, in that He wasn't trusting HIs flesh to make decisions - rather, He was putting HIs full trust on His "Father part" - since it was His decision making part, and also wasn't wearing a fleshly body.
We humans have decision-making parts too - our minds and our hearts. Without them, we would be completely idle and useless. Our minds make decisions, and tell ourselves what to do in daily life.
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