Pretty much everything Nilum said is exactly right. 07th Expansion and Type-Moon are just a lot of good. The requirements to play Battle Moon Wars and actually knowing everybody is pretty time-consuming and steep though. Fate/stay night, Fate/hollow ataraxia, Tsukihime, Kagetsu Tohya, Kara no Kyoukai, and the Melty Bloods. Thankfully Fate/zero isn't really needed.
I'd also recommend Umineko and all the many fan games for Umineko and Ace Attorney. Quality can vary but I can recommend a few good ones.
Ace Attorney:
The Contempt of Court - has two cases right now, uses PyWright and they have really good production values. Third case WIP
Ace Attorney: Judgement Seekers - two cases right now, uses Ace Attorney Online. First case isn't too too impressive but the second case is much cooler. Third case WIP
Umineko:
Witches & Woodlands - requires knowledge of at least the first 6 episodes I think? (might be 8), very very fun and feels authentic. Basically a D&D parody but Seacats. Has three parts. Part 2 and 3 are great.
I am currently playing One Hell of a Turnabout and When The Seacats Cry, but I shouldn't give recommendations yet.
I also highly recommend a few Alicesoft games. Daibanchou and Sengoku Rance are both 100% translated and quite fun. Although I haven't finished Daibanchou yet and SR is much more accessible. Daiteikoku is only 35% translated so I don't recommend it until you at least do the others.
Sadly I can't recommend the KID games Ever17 and Remember11 too much. Ever17 has a great ending but it's very boring (then again if you liked Katawa Shoujo maybe you'll like that?). Remember11 is pretty good overall but it suffers from huge budget issues and if you know MGS2 the ending suffers
HARDI think that's all I have to say for now. Listen to Nilum.
Oh.
Never
ever ever play Tsukihime first. That's a bad first VN, I'd say Ever17 is too even.
EDIT:
Higanbana completely slipped my mind. If you liked Higurashi and Umineko it's pretty much a given since 07th Expansion. I thought it was pretty good. Great music as always, some good chapters too. Only the first game is translated so far.
Aselia the Eternal (Think Ogre Battle only it takes you 3-4 hours to get through plot to the first battle and then the battles kick your ass)
Have you even finished that? I have like 14 hours and I'm only on chapter 2 and nothing has really happened. I mean it's decent at best but we need faster battles and MORE PLOT not trivial interactions.