I also started watching more anime during high school and I've definitely accumulated a few favorites, though I still haven't watched many of the classics which I plan on rectifying soon.
Angel Beats! takes place in the afterlife and it was pretty interesting though somewhat rushed towards the end of its 13 episodes. Silver Spoon is by the the same mangaka who did Fullmetal Alchemist, and the characters are all just as lovable. It takes place in a completely different setting though and focuses on a guy who decides to go to an agricultural high school. There are two seasons of it, and I definitely think it gets better as it goes on. Noragami is about a minor god trying to get people to worship him, so has supernatural elements, and the relationships of the main characters are wonderful. There's also a second season entitled Noragami Aragoto. Your Lie in April is a real tearjerker but it's told really well and the characters are great. It focuses on a piano player regaining the strength to play again after avoiding it for two years. Soul Eater is about people who can change their bodies into weapons and the main characters all attend school to improve their skills. It was one of the first anime I ever watched so I have some fond memories. Most of these are available on Netflix and all of them are on Crunchyroll.
Lastly, I'm going to be one of those people who recommends something they haven't finished yet, but March Comes in Like a Lion is honestly so good. The plot is fairly standard; the main character is a professional shogi player who lives on his own at the age of 17 and would be a loner if not for this family of three sisters who give him a place to call home. The music and art is absolutely beautiful and I can't help but love all of the characters. It also has two seasons out so far, but I haven't finished the second.