I've watch a couple of anime my friend introduced me to, in addition to a few I decided to watch myself. These include:
Nichijou. This was my first anime, and I really enjoyed its over exaggeration of everyday interactions.
Tari Tari. I don't remember this one incredibly well, but overall it was good, with some very touching moments.
Fairy Tail. This is by far the longest anime I've watched, and it's still so much shorter than One Piece or Naruto. It's quality can vary a lot, but I really enjoy fantasy, so usually it's worth it.
Sword Art Online. I loved the premise and first arc of Sword Art Online, but it has some annoying tendencies, and immediately launches into a horrible second arc.
Yona of the Dawn. This is the one I've most recently watched, and I really enjoyed it overall. Yona starts the season as a spoiled princess who's literally almost never ventured outside of the castle walls, and eventually matures into a compassionate woman who goes to great lengths to protect the people around her.