This Psalms teaches that God is Supreme, he's the most high God, He love us, he's the Majestic and is great.
Psalm 8 also teaches us that Human Being are precious in the sight of God and because he love us, he die just for our sins.
Psalsm 8 also teaches us that Jesus became human, just a little lower than the Heavenly beings and He will raise all who belong to Him above the Heavenly beings when He comes to reign over the new Heaven and new earth, Jesus is the only Person who perfectly reflects God's image( Galatians 2:20, Colossians 1:15. )
Finally it teaches us that we are responsible to serve as ruler over everything that the Lord made, and because he made us in his image and likeness, we have so much authority as children of His.