The first Godard film I saw was Sympathy for the Devil. It played at a midnight show, and lots of rockers and stoners went because it featured the Rolling Stones. By the half-way point, most of the audience was gone. It features the Stones rehearsing and recording the title song, over and over, cross-cut with gun-toting Marxist revolutionaries spouting incendiary slogans. This film came out in 1968, the beginning of Godard's political phase, in which aesthetic concerns took a back seat to left-wing political statements.
My personal favorite by Godard is My Life to Live. It is instructive to compare the looseness of Breathless with the rigidness of My Life to Live.
I would recommend Pierrot le Fou, Alphaville, Les Carabiniers, Breathless, Band of Outsiders and Contempt. You can't go wrong with any of them. Le Petit Soldat is highly political. Weekend is one of his best, but isn't on DVD. Every Man For Himself is cool (his 1980 comeback film), but it's not on DVD either.