I'm actually a Marvel fan, so sue me on my bias, but The Ultimate series aren't bad for comic newbies. Also, though it might be a little advanced, Marvel Zombies is royally amazing! :P
Personally, I am a huge fan of Fables. It's not a superhero comic book, it's about fantasy characters such as Snow White and the Big Bad Wolf who all live together in a compound in NYC. It's funny and smart and generally really cool.
Do you watch Buffy at all? I've gotten into comics, by following the continuation of the series in comic book form and I really liked them. If you like the show you should check out the Buffy Comics
I love the classics as far as comic books go. I would either start with Batman or with X-Men. In both of these situations you will have excellent storylines and will have a hard time running out of reading material
The Watchmen is a classic comic book series that is almost universally beloved. If you go to Barnes and Noble you can buy a book which includes the entire series. This is my recommendation for a good comic book to read if you are new to comics.
The question actually specifies "no violence or gore" and you suggest these series? add a comment