Old Covenant vs. New Covenant

How do the old and new covenants relate to salvation, if at all? I read in Luke 22:20 that the new covenant was in Jesus’ blood which is poured out for us. I also read in Heb 8:6-13 that the new covenant is superior to the old covenant and that He has made the old covenant obsolete. What I can’t seem to find anywhere in Scripture is how the old covenant related to salvation. The old covenant was instituted by God through Moses. Prior to Moses, Abraham believed the LORD and it was credited to him as righteousness (Gen 15:6). So here is an example in the old testament where salvation was by faith in the old testament, but the old covenant was not in effect yet. Does the old covenant speak of salvation/condemnation at all? It seems when reading Deuteronomy 30, that it is primarily about obedience rather than faith. Did the new covenant change the nature of salvation?  Now one thing I can glean from the Old Testament is the sacrificial system that is meant to atone for sins.  There was continual sacrificing of animals to atone for the transgression of laws, since no one could perfectly keep God’s laws.