Ah, well. Welcome to the forums, first of all. And a quick fix.
You're talking about Jarl Balgruuf, yea? He's a Jarl, which is kind of like... a leader who watches over the district. In this case, the Whiterun hold. The "king" of Skyrim is dead, High King Torygg, who was killed by Ulfric Stormcloak and is the reason there is a civil war going on.
First thing's first, are you on PC or a console? If PC, changing armor is a breeze.
Secondly, having a follower is nice, but Lydia is kind of one of those obligatory characters where she's gonna get in your way if you're trying to be sneaky. Not good for stealth types, but better for warrior types. There are plenty of them for hire around Skyrim, but cost about 500 gold. So, either way, it's up to you.
Third, weapons and armor typically show up on a character depending on their stats. Lydia is a heavy armor kind of gal, so if you give her leather armor over what she already has (steel, I think), then she's going to keep the steel, as that's a heavy armor class, whereas leather is light. In this case, if you were to give her Nordic Armor (which is leveled, you won't get it right now), she'd wear that.
But since you're new to the game and just starting off, you're going to have to work your way up to get the both of you some decent armor.
Remember, heavy armor makes more noise than light armor does, so not ideal for sneaky types if that's what you're going for. If you need any help, just lemme know, okay?
@Tarmiricmi
