most of the design will depend on the parts you have. The basic movments could be done with a single motor. Like the Starwars quad. In this; say the front left and rear right move then the others this gate keeps the center of gravity in check(trot). More advanced gates will need weight shifting to keep gravity in check. There so many ways to do these, a look at YouTube will give you tons of ideas, also brickshelf.
That would be alot harder to replicate in Lego than it looks. It looks like 16 dof. If the OP is looking to build their first quad 16 dof will need alot of parts and could get rather expensive as well. But it is a cool bot.
I would like to see more people post what they are working on as well, seems we are not getting many of those projects in the works stuff, would be cool to see even if they dont work.
Doc
"Anyone who has never made a mistake has never tried anything new."
Albert Einstein