I think the logical starting point for Supergirl Week is with her first appearance in Action Comics #252. I think Superman's initial treatment of Supergirl is one of the greatest (fictional) injustices of all time. Let's have a look:
Superman is just about to smash the missile to bits until he realizes that there's a person inside:
Ok, Superman in that third panel? Amazing. He goes from being totally shocked in the previous panel to being completely calm and thoughtful. And skeptical.
"I'm from Krypton!"
Man, Supergirl is just so cute in that last panel. Seriously. And I love Superman's off-panel babbling.
Supergirl cheerfully proceeds to tell the tragic tale of her family's short-lived escape from Krypton. And when she's done, Superman is shocked to learn something else:
How quickly he pulls his hand away.
So this is pretty great, right? Superman, who has been alone on this Earth for so long with no family and believing he is the last of his race learns he has a cousin! And she's here!
Yay! This will be perfect! She can live with Superman and they can bond and he'll help her adjust to life on Earth after the tragic and sudden loss of her family and equally-traumatizing solitary voyage through space!
"Yeaaaahhhhh...about that...see, I have a really small apartment, and y'know...I'm pretty busy. I can't really take care of a kid right now...I've got a lot on my plate...and the place is a mess, really..."
But don't worry! Superman has a "great idea" for her life on Earth! It's not like he's a douche or anything...
WHAT?! That is not a "great idea," Superman. That is actually the worst idea.
SERIOUSLY?! An ORPHANAGE?! After all she's been through?! What do you credit your heroism to, Superman? I'll tell you: your loving parents who adopted you when you crashed to Earth. So why do you think this is ok?
Superman gets some clothes to make his cousin less hot. He feels much better now.
Hey, you know what a good Earth girl name is? Kara.
Let's see what name she picks:
DC doesn't think much of its readers. They remind us that Linda Lee is Supergirl in the very next panel.
I also want to comment on Superman's story: wouldn't the guy at the orphanage want more information about that "big disaster that wiped out her whole community"? Wouldn't something like that have made the news?
Poor, poor Supergirl. She probably thinks this is how it goes on Earth. "Sure thing, Superman. Even though I am just as powerful as you, I will live in this godforsaken orphanage and remain unseen until you need me. Just let me know! I'll be right here...wearing this wig...alone..."
Her first Super Job? Cleaning her room!!!
Man that sucks.
No, Supergirl. No. You can't use your x-ray vision for spying on other kids. Also: you are disobeying Superman, actually, and I approve.
Proud of your cousin, eh? Maybe you should read some Superboy comics. He isn't the greatest friend in the world. You should talk to a guy named Mon-El.
Oh, Supergirl. Just you wait! I got two words for you: Flying Cat!