I haven't played the game, but here's what I know.
Byleth is the player charecters for FE:3H Byleth is a mercenary, who, after traveling for a while, is hired to be teacher at a school for nobles, which contains a church where his dad just so happens to work. He has to pick one of the 3 houses to pick. You may have seen a trio of charecters that people wanted instead, in either yellow, red, or blue? They are the leaders of the three houses. There's a timeskip, and the 3 kingdoms that correlate to the houses are at war. Not sure who the elf girl is exactly, but she is always in the back of Byleth's mind, and she has some importance to the plot. They were at her throne. No idea who the old man is. The sword/whip thing I don't get either.
You gotta remember, with FE, not every game plays into the next. Every few games or so, they move to a new continent. So, Marth has his own 2 or 3 games, Ike has a couple, roy has a couple. Chrom, Lucina (Chrom's daughter), and Robin all have their own game, Corrin has his/her own, (which kinda bleeds into Chrom's).