The Outer Limits, Season 1 (1963)

"There is nothing wrong with your television set.
Do not attempt to adjust the picture. We are controlling transmission.
If we wish to make it louder, we will bring up the volume.
If we wish to make it softer, we will tune it to a whisper.
We will control the horizontal. We will control the vertical. 
We can roll the image, make it flutter. We can change the focus to a soft blur, or sharpen it to crystal clarity. 
For the next hour, sit quietly, and we will control all that you see and hear. 
We repeat: there is nothing wrong with your television set. You are about to participate in a great adventure. You are about to experience the awe and mystery which reaches from the inner mind to... The Outer Limits."

This evocative opening narration graces every episode of The Outer Limits, an anthology science fiction show that launched in 1963.  The debt that it owes to The Twilight Zone is thus pretty clear, even before you've seen a single minute of footage, but it would be unfair to dismiss it as "just" a clone of the Zone.  The Outer Limits pursues a much more explicitly science fiction themed agenda, with most of the stories either involving an encounter with some sort of alien creature, or the development of a startling new technology and what that means for humanity.  Each episode also runs longer than most seasons of the earlier show did (though ironically, 1963 was also the one year where The Twilight Zone experimented with 50 minute, rather than 25 minute, episodes).  This extra length is not always a positive, I have to admit: several of the stories here would I think benefit from being shorter, as the pace of the show can sometimes be rather on the slow side.

If you're interested in a TV show that's canted more toward the 'science' end of science fiction, and are willing to accept that its age means it is sometimes naive or out of date in its views, then you should probably give The Outer Limits a look.  Double-check whether you're getting the original or the 1990s relaunch, though - the latter is a somewhat different beast, as best I recall it.