Two titles I would definitely recommend to people new to anime are Shouwa Genroku Rakugo Shinjuu and Boku dake ga Inai Machi. I believe they lack the various factors that might discourage newcomers from watching any more anime, and might also be appealing to people more used to western movies and TV shows. I would also throw in Kokoro Connect and Hyouka for good measure, though it's been a while and do not recall how strong they start off or maintain the pace—and do note that these two also have a fair amount of Slice of Life, which can get a bit boring for some.
Another thing that might be in favour of Rakugo would be that it is a Josei anime, meaning that it is aimed at women in their late teens on into adulthood (though as a man, I have to say it is one of my favourite anime overall, so the target audience shouldn't necessarily be taken to heart). I'm not sure how young your sister is or what she's like, but I believe that she might enjoy it better due to that. Another Josei anime I would definitely recommend is Sakamichi no Apollon, though I think it has much slower pacing than Rakugo. Alternatively, for even younger women (Shoujo anime), I recall I found Tonari no Kaibutsu-kun fairly enjoyable.
Additionally, you could always consider showing her some movies such as Kimi no Na wa or Toki wo Kakeru Shoujo.