The decade of the 1960s started off rather slow, compared to the pace set by the 50s. Some of 1960's sci-fi films had been produced in '59, so might be considered 50s films in a way. The themes and quality did not change very much at first, but would become quite different by the end of the decade. That's why the image of the car -- The Chevrolet Corvair is a sort of poster child for America in the 60s.
Here are the sci-fi films of 1960 in roughly chronological order:
Angry Red Planet -- Yet another first manned mission to Mars in the mode of Rocketship X-M. The crew encounter troubles and are told they're not welcome on Mars.
Visit To A Small Planet -- Jerry Lewis plays a mischievous alien who comes to study earth, but finds he cannot control himself.
Leech Woman -- An unscrupulous and abusive scientist learns of a fountain of youth drug guarded by a jungle tribe. His wife takes it, but must kill men to stay young. Her husband was the first to die.
Battle In Outer Space -- A grand epic from Toho Studios, set as a loose sequel to The Mysterians. Dubbed into English, earth scientists must defeat the Natalians who wish to make earth a colony.
12 To The Moon -- A cosmopolitan crew make the first manned moon mission, only to discover that advanced lunar beings don't want earthlings there. Earth is almost doomed until human love softens the lunar rage.
The Lost World -- Irwin Allen's remake of the 1925 version of Doyle's novel -- the prototype for Jurassic Park. Budget constraints kept the movie from greatness.
The Amazing Transparent Man -- A mix Invisible Man and a film-noir crime drama. A shady spy forces a German doctor to make a criminal invisible in order to steal funds and isotopes to create an invisible army.
Beyond the Time Barrier -- A test pilot flies through a time displacement to arrive in 2069. Mankind is doomed to mutation from a nuclear-induced plague. He must return to 1960 to warn everyone.
The Time Machine -- H.G. Wells' novel is given A-level rendition in this classic film.
Dinosaurus -- A construction crew on a caribbean island unearth two "frozen" dinosaurs and a cave man. A lightning storm awakens them to mayhem and adventures and a duel between T-Rex and a steam shovel.
The Cape Canaveral Monsters -- Alien life forms (dots of light) inhabit two earth scientists who then set about sabotaging America's space race efforts. A clever hero outwits them, or does he?
Nebo Zovyot -- This high-budget Russian space adventure rivals Conquest of Space ('55) in tone and effects. Never released as a simple English dubbed film, footage was used in later American B films.