If you were to give the writers of Supergirl some ideas for the show, what would they be? Here of mine.

Introduce Superman

I say introduce Superman, but not immediately! There is no need. If anyone has read a decent number of Supergirl comics, they would know that her stories barely involve Superman, and that Supergirl can more than take care of herself without Superman's help. However, she does hail from the same planet and family as Superman (cousins), so it would make sense to introduce him. But maybe at the end of Season 1 or Season 2.

Introduce other young female characters

Introduce other young female heroes, no sense in letting the Teen Titans TV series hog them all. Here are a few good ones.

Batgirl: Batgirl and Supergirl, for some of the female comic book fans, this would be the true Dawn of Justice. We don't have to introduce Batman or Nightwing (Batman/Bruce Wayne is owned by WB, and Bruce Wayne is owned by Fox), it won't be all that necessary. As far as the viewer can be concerned. Barbra is the only vigilante defending Gotham and does a damn good job at it.

Miss Martian: It would make sense to make for Miss Martian and Supergirl's human personas to somehow meet, become friends, only to encounter some kind of threat that forces them to reveal their powers to each other. Then they can build an even more dynamic friendship since both of them are extraterrestrials on Earth who are making their best to fit in, while also using their powers to protect innocent people.

Mia Dearden Speedy: If due to Fox or WB (Batgirl might exist in the Snyderverse, and technically Barbra Gordon just doesn't exist quite yet in Gotham), then they can just do what Smallville did and use the next best thing, use an archer character. Technically Thea Queen's full name on Arrow is Thea "Mia" Dearden Queen. So they may or may not be able to use her, which is why I suggest Artemis Crock.

Wondergirl: These two teaming up and kicking ass, fighting either either Wondergirl or Supergirl villains. No further explanation required, it'll be awesome.

Why introduce superman? For what? They are better off honestly just sort of ripping off the Superman story and making her female Superman. I'm not even joking.

I want the Kara I know and love, the rebellious anti-authority, anti-hero, anti-Superman.

All the girls you mentioned are Teen Titans except Barbara and she's been definitely confirmed for that show already. I'd like Supergirl to explore the more cosmic qualities of the DC universe, the Martians is one suggestion I definitely agree with, J'onn J'onzz has to belong somewhere onscreen if the DCCU isn't interested in using him, he could be a cool mentor-esque figure and their personality clash could be very entertaining imo.

Here's hoping Supergirl, Titans, Arrow, Flash and whatever-the-new-CW-show-will-be-called are in the same universe; various travelling/visiting characters could keep all these shows fresh for a long time.

spend the first season or so on krypton

@black_wreath:

Barbara and she's been definitely confirmed for that show already.

What did I miss?

@rbt:

Barbara Gordon will be Oracle on Titans, it was announced earlier this year.

so many tv shows

@masterkungfu: no, they should just have flashbacks to Krypton like Arrow has flashbacks to Lian Yu.