This will be part I of a three part Lineup analysis, focusing on shows in the upcoming season that I'm DEFINITELY going to watch.
I found this chart at http://www.alafista.com/2013/04/25/summer-2013-anime-lineup/ and if this chart can be trusted, there are a couple shows that I'd be willing to check out, though nothing I'm really thrilled about as of yet. A lot of the shows I can't even watch because they're continuations of ones I didn't watch in the first place.
Honestly, the show I'm most geared up to see is Kamisama no Inai Nichiyoubi, which I think translates to "The God of Lost Sunday" or "The Sunday when God Disappeared" (really not my best work >__<;;). Whatever, it has something to do with God not existing and Sunday. It's got an interesting premise, and from what I can get from the thumbnail, it's got a pretty solid art style, and I think Madhouse did a show in the past that I really enjoyed...though the name escapes me.
Secondly, I'd be down with watching Tokuei Sochi Dantai Stella Jo-Gakuin Koutou-ka C3-Bu which I'm not even going to attempt to translate. For starters, it's directed by GAINAX, out lord and savior. Second, the premise reminds me a bit of K-On!, a series that I thoroughly enjoyed.
Lastly, I think I'm going to watch the two shows directed by Brains Base: Blood Lad and Brothers Conflict. They came at the suggestion of my fiancee, who has never steered me wrong before. Furthermore, the aforementioned director is also shared with Durarara!! and Baccano! both shows I have yet to finish, but get a solid recommendation nonetheless.
Also, I'm going to be continuing Attack on Titan and Suisei no Gargantia, which I'm fairly certain are going to be continuing into Summer.