Here are a few recommendations of Shojo (aka Shoujo, or Anime for girls/women) "slice of life" Anime are:
Nana, Lovely Complex, Clannad, Clannad After-Story, Chobits (if you like sci-fi), Hibike! Euphonium, Elfen Lied (sci-fi, with graphic violence), Marmalade Boy, Nodame Cantabile, Fruits Basket, Gunslinger Girl, and the saddest story of all time, Your Lie In April.
If you like Classical music, watch Nodame Cantabile, Your Lie In April, and to a lesser extent Hibike! Euphonium (Sound! Euphonium).
All of these are easily found on free sites all over the Weeb.