Any specific recommendations for a teenage girl just getting into anime sans moe, slice of life, and anything too intense or with horror elements?
It would probably help if we had a bit more info, but I'm guessing given you're posting in this thread that you're not interested in anything with heavy fanservice and into shows with good female characters? And also appropriate for younger viewers (although teenage is a pretty broad age range)?
In that case, my general starter anime recommendations would be:
With female characters in leading roles
Kaguya-sama: Love is War - Great romantic comedy, with some light fanservice scenes that were added in the anime adaptation, but nothing too terrible.
Monthly Girls' Nozaki-kun - Another great romantic comedy that's pretty universally praised. Although this one feels like it benefits from having a knowledge of some anime tropes for certain jokes.
Little Witch Academia - A pretty generic choice honestly, but fits very well with what you're looking for.
Shirobako - A show about the process of making anime, featuring cameo appearances from a large number of actual industry professionals. Both educational and also really well made. But it's also slice of life, so maybe not? And given that it deals with adults in the workplace, it's aimed at an older audience. So I don't know how much younger viewers will like it. I guess it would depend heavily on their specific age.
A Place Further Than the Universe - A really good drama about a group of girls making a trip to Antarctica. But it might be too moe or slice-of-life for you, depending on what those word even means to you.
Sound Euphonium - Girls playing instruments in a music band, with really beautiful visuals. Although from a feminist point of view it comes across as pretty problematic to me, and the second season of it really sucks.
K-On - You said no moe and slice of life, but I'm not 100% sure why given that shows like K-On are literally made exactly for teenage girls. Anyway, it's a good show that gets a ton of hate from people who've never even seen it just because of its artstyle. I actually avoided watching it for a long time because of its reputation, but ended up really liking it.
Without female characters in leading roles
Hikaru's Go - I think this is pretty much the best generic shounen anime out there. And unlike most of them, it's about a kid playing a board game, so there's no violence involved.
The Eccentric Family - The closest an anime has ever come to making me feel like I'm reading a good YA book. With really nice visuals, and a relaxing urban fantasy story about a family of shape-shifting raccoons living in modern day Kyoto.
Haikyuu - Pretty much the best standard sports show you can get. Although maybe it's too intense? There's no violence or anything like that since it's just sports, but there's definitely a large level of intensity to it.
