A family of four, dad, mom, son, daughter and their two strange pets are traveling through space in search of strange new worlds.
I think you are talking about the Mezga Family an hungarian cartoon. Different names in different languages, but check the series here: http://en.wikipedia.org/wiki/The_M%C3%A9zga_Family
check this italian website for some pics: http://www.anni80.info/cartoni/mezzil.html
The cartoon that immediately came to my mind is "Les Mondes Engloutis" (English: "Spartakus and the Sun Beneath the Sea", http://en.wikipedia.org/wiki/Spartakus_and_the_Sun_Beneath_the_Sea).
The story is about a weird girl, a man with a cool bracer, 2 kids and 2 strange, tatoo like pets travelling on a rubber duck-shaped spaceship-like vessel called ShagShag from their city located inside the planetary crust of the Earth towards the surface with the goal to save their artificial sun from dying.