The Apostle Paul explains that through faith in Jesus Christ every person around the world will have the opportunity to be saved by obeying the Gospel of Jesus Christ:
Now to him that is of power to stablish you according to my gospel, and the preaching of Jesus Christ, according to the revelation of the mystery, which was kept secret since the world began, But now is made manifest, and by the scriptures of the prophets, according to the commandment of the everlasting God, made known to all nations for the obedience of faith (Romans 16:25-26).
After Paul's "road to Damascus" experience (Acts 9: 3-9), he was filled with the Holy Ghost and was baptized at Ananias' house (Acts 9: 17-18). Having obeyed the gospel - being born again, he was a new creature as Paul wrote in 2 Corinthians 5:17. He clearly demonstrated this, almost immediately after he was saved, by preaching Christ (Acts 9:19-22) instead of persecuting Christians.
Paul says that he received the gospel, not by man but by the revelation of Jesus Christ:
But I certify you, brethren, that the gospel which was preached of me is not after man. For I neither received it of man, neither was I taught it, but by the revelation of Jesus Christ (Galatians 1:11,12).
The Bible is not clear about the exact timing when Paul received the gospel by the revelation of Jesus Christ, but it had to happen sometime between the time he met Jesus on the road to Damascus to when he started preaching Christ, or otherwise how would he be able to preach the gospel in Acts 9:20.
Finally, the Bible instructs us that all scripture comes from the breath of God, the Holy Ghost:
All Scripture is breathed out by God and profitable for teaching, for reproof, for correction, and for training in righteousness (2 Timothy 3:16).
The conclusion here is that when Paul says "obedience to faith" we know that from his own personal experience i.e. being born again and receiving the revelation of Jesus Christ; and from the Holy Ghost being upon the Apostle Luke when he wrote the Book of Acts, that every person around the world, through faith, will have the opportunity to obey the gospel of Jesus Christ to be saved, to obtain the salvation that affords us the opportunity to enter into the Kingdom of God.
When you look at Strong's definition of save it is what one would expect to see with regards to what Jesus does for us - save, deliver, protect, heal, preserve, do well, make whole:
Save - sṓzō( sode'-zo) G4982 from a primary σῶς sōs (contraction for obsolete σάος sáos, "safe"); to save, i.e. deliver or protect (literally or figuratively):—heal, preserve, save (self), do well, be (make) whole.
Having said that though, it doesn't specifically say "through Christ" but when you look at the outline of Biblical usage in the definition it does say "to deliver from the penalties of the Messianic judgement", or in other words "to get to heaven".
From the definition of faith, it is clear that faith is about the "moral conviction of the truthfulness of God, especially upon reliance on Him for salvation":
Faith - pistis (pē'-stēs) G4102 - from G3982; persuasion, i.e. credence; moral conviction (of religious truth, or the truthfulness of God or a religious teacher), especially reliance upon Christ for salvation; abstractly, constancy in such profession; by extension, the system of religious (Gospel) truth itself:—assurance, belief, believe, faith, fidelity.
That definition says pistis (G4102) comes from G3982 - persuasion, so let's look at the definition of persuasion:
Persuasion - peithō (pā'-thō) (G3982) is to convince (by argument, true or false); by analogy, to pacify or conciliate (by other fair means); reflexively or passively, to assent (to evidence or authority), to rely (by inward certainty):—agree, assure, believe, have confidence, be (wax) conflent, make friend, obey, persuade, trust, yield.
As you can see, peithō also means obey and since the word faith comes from persuade (or obey), obedience cannot be separated from faith. In other words it follows that one cannot have faith in Jesus, without obeying him.
That question is best answered by looking at the words directly from the mouth of Jesus. After Christ was risen, he compared being faithless to being unbelieving when Thomas did not believe that Jesus had rose from the dead (John 20:25). Jesus said to Thomas:
Reach hither thy finger, and behold my hands; and reach hither thy hand, and thrust it into my side: and be not faithless, but believing (John 20:27).
Of course, what Jesus said to Thomas lines up with the definition of believe, in that faith and believe are almost one in the same:
Believe - pisteuō (pist-yoo'-o) G4100) - from G4102; to have faith (in, upon, or with respect to, a person or thing), i.e. credit; by implication, to entrust (especially one's spiritual well-being to Christ):—believe(-r), commit (to trust), put in trust wit.
This definition says "believe" comes from the word faith. The definition of faith pistis (above) states that it means belief or believe. It then follows that obedience is definitely the action that comes out of believing as well.
Jesus is very clear about what He expects of believers, what their obedience with regards to salvation would look like to Him:
I say that because the death, burial and resurrection is the gospel (1 Corinthians 15:1-4) and the three points above line up perfectly with what the Apostle Peter preached (Acts 2:38), the day of Pentecost, the birth of the Church. This is not surprising because the risen Christ spent forty days with the eleven remaining apostles, opening their understanding of the things of God (Acts 1:3).
If we have faith in the Lord Jesus, we will believe in Him causing us to act on what we believe, causing us to obey Him. He says that:
If you love me, keep my commandments (John 14:15).
By being obedient to Jesus Christ, our actions will show that we are faithful and that we love Him. Not only that, obedience to the Gospel of Jesus Christ will cause us to be born again, opening the door to heaven.
The meaning of "faith in God" is easily understood by reading the first verse of the so called Heroes of Faith chapter in the Book of Hebrews:
Now faith is the substance of things hoped for, the evidence of things not seen (Hebrews 11:1).
The Living Bible sheds additional light on this verse:
What is faith? It is the confident assurance that something we want is going to happen. It is the certainty that what we hope for is waiting for us, even though we cannot see it up ahead.
Let us look at some other scriptures to see the benefits in having faith in God.