Depends on the genre you like...
Romance: Toradora, Tonari no Kaibutsu-kun, Ouran Highschool Host Club, Peach Girl
Thrillers/blood gore: Deadman Wonderland, Mirai Nikki
Shonen(for boys): Death Note, Bleach, Fairy Tail, One Piece....
There is soooo much to choose from!
Hitman Reborn
Bleach
Naruto
Fairy Tail
One Piece
Kaze No Stigma
Fruits Basket
Ultra Maniac
Dn Angel
Clannad
Angel Beat
Sword Art Online
Death Note
Code Geass
Black Butler
Ouran High School
and many more ....
My two favourite animes are Naruto and Bleach. Definetly try them out, especially the first seasons of Naruto. Don't start with Naruto Shippuden right away. Naruto Shipuden already is a big mess, but even more so if you haven't got an understanding of the original story. The beginning of Naruto has a really compelling story and characters. Definetly try out Naruto. I'd advise you to stay away from Shippuden.
-Fooly Cooly (A classic!)
-Deadman Wonderland
-Digimon (another classic!)
-Hetalia
-Bleach (You're gonna have to skip the filler arcs, though. *shivers* So much filler...)
-Subbed-YuGiOh
-Ouran High School Host Club
-Casshern Sin
-Full Metal Alchemist
-Eureka 7
-Soul Eater (I haven't watched it yet, but I heard that it was good!)
-Ghost In The Shell
-Ai No Kusabi (yaoi)
-Descendants of Darkness (shounen-ai)
There'es alot of genres in anime so I would ask you what kind of genre (Romance,action,sci-fi,etc) you like from other media and then suggest you a some anime to watch.
Also I don't know why so many people hate fillers. (Fillers are episodes that pretty much are the episodes that aren't apart of the plot and sometimes the fillers are actually good. Depending on what anime you watch.)
Action (and alot of other things) : Bleach,Naruto,One Piece, Fairy Tail, Code Geass, Soul Eater
Romance: Toradora, Ouran High school host club
Suspense and Horror: Death Note, Deadman Wonderland, Mirai Nikki. (The last two animes are kinda bloody.)
And there's others. There are literally tons of anime to choose from. From obscure to extremely popular.
sad anime for you: clannad
happy anime for you: school rumble, k-on, hetalia
gory anime for you: mirai nikki
magical anime: Puella Magi Madoka Magica
again, i really recommend k-on. it's a really cheery friendship anime which is my favourite.