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Some animes appeal to a specific gender/age and are hit and miss, but I am surprised to see those presented as for general audiences.
For example, I would like to recommend Robotech but it has a retro feeling to it, or Mobile Suit Gundam Wing but is a long plot which needs patience like Death Note.
On the other hand, you have good plots with occasional lighthearted stories like Slayers, although still has a retro feeling.
Now, good stories with amazing fiction (and good fighting scenes too) are those who you can call entry level. I strongly suggest go for Vampire Hunter D: Bloodlust, which may seem not fit for aficionados, but is just enough to show there is more to anime than niche stuff. But when you want full anime, Rurouni Kenshin for one of the best masterpieces out there, even the music score is something beautiful, although it has comedy moments you can well understand comedy relief in Japan is particularly innocent, but when the drama appears you intermediately immerse in it.