For to this end we both labor and suffer reproach, because we trust in the living God, who is the Savior of all men, especially of those who believe.
1 Timothy 4:10 (NKJV)
One of the things I love about this verse is the fact that Paul, speaking of God's salvation, addresses both believers and unbelievers when he says that God is the savior of all men. Several passages in the Bible actually show that God's salvation is for those who believe. This makes many people, including Christians, believe that the Bible focuses more on the doctrine of salvation, forgiveness and sanctification, which concerns Christians.
It is true that when Paul says that we have put our hope in the living God in this verse, he is speaking only to those who have already believed in Jesus Christ. However, he does not limit salvation to believers only. By saying that God is the savior of all men, he clearly shows that God wants “all men to be saved and to come to the knowledge of the truth.” - 1 Tim. 2:4. Salvation being for all. You just need to receive it by faith.
If God has given grace to all men, he nevertheless leaves them the right to accept or refuse it. But only those who believe will enjoy salvation and the blessings that come with it. Do not be one of those who look at God's salvation from afar, but open your heart and receive with joy the grace of God which leads to salvation.
May the grace and the peace of God be with you all.