I started watching Supergirl from the pilot when the show had big ambitions and was going to air on CBS. The pilot of the show is... not great. You could feel the hands of CBS, home to a wide array of boring shows I don't watch, all over the show and kind of wringing themselves with all this superhero weirdness. But they did bring in a decent cast, and seemed to have some ideas for modernizing the Maid of Might from her incredibly goofy origins in Action Comics 252.
I won't get into it here, but Supergirl as a property allows for some flexibility as the character's titles never last, no one working on the latest iterations seems aware of prior incarnations, and once on the title, never seems to know what to do with the character for more than 3-6 issues. I have probably hundreds of Supergirl comics, and there's been exactly two modern runs that I would recommend.
The show started off on CBS, which was always an awkward fit and probably one of the things that drove DC and WB to realize that working with network suits was more trouble than it's worth. For the first half of the first season, the show felt deeply uncomfortable with itself, bucking against old network tropes and trying to make the domestic life of Kara Zor-El as basic as possible.