Reading through this, I have to comment on the difference in attitudes toward anime in and out of Japan.
Western fans tend to like
anime. Japanese fans tend to like
specific shows.
I have used this example countless times, but it`s kind of like, well, liking magazines. Anime is a media form - not a genre. Saying "I like anime" is along the lines of saying "I like magazines!"... Or "I like television programs!" It is incredibly broad, and sounds kind of stupid when you think about it.
But for western fans, it seems that anime is a genre. "If it`s anime, I know I`ll like it" sort of thing.
----------
Back to the suggestions though.
Instead of trying to pick up and watch a show with the intention of finding some anime you like.... Why not do what most Japanese (who watch anime "casually") do?
Look at the
listing for the upcoming/current season of anime, see if you can spot something that doesn`t look unappealing, check out what kind of story it has on wiki... And if it sounds like it might be interesting, try watching the first episode or two. If you think it`s good, keep watching. If you don`t, stop.
For the summer lineup, we`ll be continuing to watch 夏目友人帳(going into the third season now?) and checking out 2 or 3 new ones that
might be decent.
It`s just like any other type of programming. If it looks decent, watch the first episode or two and see if it`s something you would want to keep watching.