Question: What is the Holy Spirit or Holy Ghost? Which is better to say? What do I need to know or do about the Holy Spirit to follow Jesus and believe correctly in God?
Response: This is an excellent question to which my response can only be one that is very basic (although I think that getting clear about basic matters is tremendously important).
First, "ghost" is simply an old translation of "spirit" into, first, German (Geist) and then into English (ghost). Therefore "spirit" and "ghost" are identical in meaning, but it is better to say "spirit" because today "ghost" brings with it other meanings, such as those in stories about ghosts and haunted houses.
Second, the word "spirit," by itself, can mean almost anything, whatever we want it to mean. People speak of team spirit, the spirit of the age, school spirit, the spirit of Lincoln, the spirit of the Confederacy, or in the case of the Nazis, the spirit of the Fűhrer (the "leader," i.e., Adolf Hitler). In theology we speak of the Spirit as the Spirit of God or the Spirit of Christ, or as the Holy Spirit, to make clear how we are using the word. We learn what the Holy Spirit is by reading the Bible.
The Hebrew and Greek terms for spirit are ruach and pneuma; each term refers to breath and/or wind. The biblical narrative is one, large sweeping testimony to the universal presence of the Spirit. God's spirit was active in creation: "A wind from God swept over the face of the waters" (Gen. 1:2). The spirit is active in maintaining human life (the metaphors for human breath and soul mean "spirit). The spirit of God acts in every situation, whether in nature or history. Extraordinary endowments for leadership, especially prophecy, and due to the spirit. When David spoke as a prophet, he claimed "The spirit of the Lord [ruach adonai] speaks through me" (2 Sam. 23:2). Second Isaiah declares: "And now the Lord God has sent me and his spirit" (Isa. 48:16). Ezekiel claims that the Spirit "falls upon him," "enters" him and tells him what to say (Ezek. 2-11).
Also, wisdom and discernment are special gifts of the Spirit. The Wisdom of Solomon beautifully expresses the close relationship between the divine Spirit and wisdom: "Who has learned your counsel, unless you have given wisdom and sent your holy spirit from on high?" (9:17). The Spirit is God active in all history, and chiefly in Israel's. It is immanent because transcendent, coming from the life of God who deals directly with human beings. Spirit stresses the nearness, the presence of God. "Where can I go from your spirit? Or where can I flee from your presence?" (Ps. 139:7). Jews later came to speak of the Spirit as the Shekhinah, the all-present one. As Rabbi Hanina b. Teradion said: "If two sit together, and words of Torah are spoken between them, the Divine Presence rests between them" (Pirke Aboth 3.2 in The Mishnah).
The Spirit is active in the life and ministry of Jesus. Jesus is conceived by the Spirit; Jesus is the gift to the world and to the church from the God of Israel and also from the Israel of God (Jesus was an Israelite). The Spirit descends upon Jesus at his baptism, his temptation, and at the critical hour when his ministry begins, to fit him for his unique vocation of being a servant to the people Israel (Rom. 15:8) in order that the Gentiles might glorify God for God's mercy (Rom. 15:9). In the synagogue at Nazareth, Jesus reads from Isaiah: "The Spirit of the Lord is upon me, because he has anointed me to bring good news to the poor" (Luke 4:18).
In the great commission, Jesus instructs his followers to make disciples of all the Gentiles, all the ethnē, i.e., "ethnics." The Spirit was active in the creation of the church at Pentecost (Acts 2).
Jesus sums it all up in John 4:24: "God is Spirit." We talk of God in many ways, because we know God in many ways. Sometimes we talk of God by calling God "spirit." God is spirit.
What we need to know is that we learn about the Spirit simply by reading the scriptures. What should we do: live according to the teachings of Jesus and the apostles. Love your neighbor as yourself and act accordingly. That will be following the spirit of Christ.