1 Peter 5:3  “Neither as being lords over God’s heritage, but being examples to the flock.”
It’s always been distasteful to me when leaders in businesses or organizations have a superiority or supremacy attitude towards those they are supposed to be leading, but more so in the church.  Our example is Christ Himself.  Although He truly was and is superior, He laid that aside and showed Himself a servant in order to serve us and to manifest His great love for us.  We must not do differently.

5  “…be subject one to another, and be clothed with humility…”
As Paul wrote, let each esteem others better than himself, look out for the affairs of others.  Employ Christ-like servanthood in dealings with others.
I’ve also heard this verse used as a counter point to Peter saying wives should submit to their own husbands.  Sad that folks get hung up on women submitting.  Perhaps it’s a reflection of incorrect feminist influence on the culture.  At any rate, Peter isn’t speaking of a marriage relationship here.  Certainly, a husband and wife should submit to each other as fellow heirs of Christ, as this context says.  But within a marriage relationship, other protocols are active.  Nothing that would establish equality of person, but of position.

6  “Humble yourselves therefore under the mighty hand of God, that he may exalt you in due time”
Not a promise of gain or prosperity.  If I submit to God, He will do this and that for me.  But a submission to His perfect will, way, and timing.  Submit to Him, and He will make us like Christ and bring glory to Himself through us in His time.

7  “Casting all your care upon him…”
Sometimes difficult to do, to let go.  So much easier to worry about stuff, especially when He seems to be taking His time about things.

9  “…knowing that the same afflictions are accomplished in your brethren that are in the world”
Not alone in our suffering.  And a reminder to pray for others who may be suffering greater trials for Christ.

12  “By Silvanus…I have written briefly…”
Appears Peter may have dictated the letter to Silvanus or had him carry the letter.  Commentary favors the latter.  Not uncommon.  Paul sometimes did so as it appears he had some vision problems.  Curious as to the reason if he did dictate.  Physical infirmity or just more convenient?

13  “The church that is at Babylon…”
Other translations render this as She who is at Babylon.  Shows the difficulty in translating between any two languages and the different methods that are used.  Commentary mentions different views.  Either one woman or the church meeting in Babylon.  It favors the latter.  Makes more sense to me also given the context.

13  “…Marcus my son”
Apparently same Mark that was unfaithful to Paul on the first missionary journey.  He redeemed himself well, with Paul and Peter, and wrote the gospel account bearing his name.  Good example of God’s mercy and grace.  He doesn’t throw us away when we make mistakes if we stay and trust Him.

I hope you enjoy reading and studying His word.  May it accomplish what He desires.  Please feel free to comment or post questions.  Thanks for reading!