A list of some basic, must-sees, either because they're amazing or incredibly popular. Or both:
- Fullmetal Alchemist: Brotherhood (My favorite anime ever )
- Puella Magi Madoka Magica (My third favorite anime ever )
- Angel Beats
- Clannad
- Code Geass
- Cowboy Bebop (Though I haven't personally seen this one)
- Anohana

Additionally, here's everything that I've ranked 10/10 on My Anime List. You can see the genres to the right:




And here's everything that I've ranked 9/10:


Of those shown in the images above, given the genres you've expressed interest in, I'd recommend:
- Aldnoah.Zero (Action)
- Blue Exorcist (Ao no Exorcist) (Action)
- Kill la Kill (Comedy and action are both in this one)
- No Game No Life (Also has some comedy and action/adventure)
- Zankyou no Terror (Action and thriller )

Lastly, some things that you should maybe steer clear of (Though Neon Genesis Evangelion is a favorite of many, it just wasn't up my ally. It's also very popular, so don't disregard it from this. Everything else listed below, I'd advise you avoid):

(This last list may get me some salt, oops)

I hope this helps!  

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EDIT:
I see that you've also expressed interest in the romance genre! In that case, try the following:
- Kaichou wa Maid-sama! (Comedy and romance, hilarious and cute)
- Monthly Girl's Nozaki-kun (Comedy and romance, has made me laugh more than any other anime)
- The Pet Girl of Sakura Hall (Comedy and romance, my second favorite anime of all time)
- Nagi no Asukara (This has a similar feeling to Anohana, and it's got some romance in it)

I know what you mean, it's about the genres for me. I like nearly all anime I watch weither films or series, saying that if \I watch alot of the school based animes they tend to get a little samey for me. Here's a few i've watched, willing to watch again or in process of watching (in no particular order.

The films
Perfect Blue
Akira
Anything from the Studio Ghibli collection (except pom Poko ~ I didn't enjoy that one)
Ghost In the Shell
Ghost In The Shell 2


The Series
Full Metal Panic
Full Metal Panic Fumiko
Naruto:Unleashed (not the cut down Nicolideon version)
Naruto:Shippenden
Claymore
Ghost In the Shell:SAC 1
Ghost In The Shell:SAC 2
Fairy Tale

I probably have others laying about, I try to add to the list later. If it's high fantasy or sci-fi I'll probably enjoy watching it.