Humor is key so I picked out a couple of comedies that may make a good point of introduction for anime noobs.
Lucky Star
First we have Lucky Star which is a natural fit for this.
It's very family friendly, no bad language, no real fanservice, and is really funny. The problem is that most of the humor comes from the references to other anime, especially Haruhi and other 80s-early 00s anime.
It's still funny even without that, but many of the jokes are a lot funnier if the context is known.
Cromartie High School
Cromartie High School is very funny and has no nudity, very little violence, and a moderately low amount of swearing if I suggest you watch it first to see if it's something that would be okay for your family though.
The Melancholy of Haruhi Suzumiya
The Melancholy of Haruhi Suzumiya is often overlooked as a comedy.
It has a fairly small amount of fanservice and is a great introduction to the rest of the world of anime.
I can only really think of a few problematic scenes that may make this an uncomfortable anime to watch. However, I will say that they are all extremely short and don't really show much.
My Bride is a Mermaid
My Bride is a Mermaid is a great rom-com with very light fanservice (again, just a couple of scenes that aren't very long at all) and has compelling characters and a decent dub.
I know that some people say that My Bride is a Mermaid's dub is bad, but I think that it is wonderful if not fantastic.
Gekkan Shoujo Nozaki-kun
Gekkan Shoujo Nozaki-kun is a fantastic fit for this, but it isn't dubbed yet. The dub was announced not too long ago. It might be worth picking up once it does get dubbed.
And to conclude our list, anything staring Crispin Freeman. He has just enough snark to make anything really a fun, humorous option that is really well dubed. Yay, Crispin!
