The period when all the great TV shows were on depends entirely on your personal circumstances. A lot of people I know rave about shows from the 80s, but I spent part of that decade without a TV, and I watched relatively little for most of the decade, so I missed out on many of these classics. I even lost touch with Doctor Who during this period.
My personal "Golden Age", both for music and TV, was undoubtedly the 60s. I began the decade in the infants school and finished it as a highly aware teenager; and, while rationally I accept that it's my bias, it seems to me that the whole era was bursting with creativity and imagination in a way that's never been rivalled since.
60s TV means various things to me — notably the era's brand of surreal comedy that culminated at the end of the decade with Python — but sci-fi looms large in it. Fantasy much less so. One or two of the US sitcoms we got included a slight fantasy element (Bewitched, for instance, in which an ordinary family sitcom had the twist that the wife was a magical being) but the only out-and-out fantasy show I recall (leaving aside talking animals and toys coming to life, which constitutes social realism for young kids) was Noggin the Nog.
Noggin was ostensibly aimed at very young children, but was one of those programmes that adults can enjoy on a whole different level. It was narrated over still illustrations, which doesn't sound promising but worked well, and captured a magical feeling right from the regular opening words: Listen, I will tell you the saga of Noggin the Nog, as it was told in days of old by the men of the Northland as they sat around their great log fires. (That was purely from memory, by the way.)
In a pseudo-Norse saga, the young king of the Nogs encounters dragons, woos a beautiful princess from a far-off people, and fights his wicked uncle, Nogbad the Bad. The whole thing's splendidly poised between serious use of fantasy tropes (such as a sword that gives its wielder absolute power) and tongue-in-cheek silliness.
In general, though, speculative TV of the 60s was sci-fi — some US imports, most of which I didn't watch, for some reason, but mainly home grown. The great legacy from the previous decade had been Quatermass. I was far too young to have seen these at the time (the first was on before I was born) but I own the 14-out-of-18 episodes that survive on DVD.
Quatermass set a very high bar for serious, grown-up sci-fi on TV. This was no gung-ho space patrol: the hero was a middle-aged scientist using scientific principles to thwart various alien invaders of earth. The first, The Quatermass Experiment in 1953, was technically primitive, but more than made up for that in storytelling, while Quatermass II (1955) and Quatermass and the Pit (1958) still look pretty good.
This tradition continued into the 60s, with serials like A for Andromeda, written by astronomer and science fiction author Fred Hoyle and featuring a young Julie Christie as the alien. I didn't see that, either, but I do just remember a serial from 1962 called The Big Pull — very dark sci-fi, which finished with Earth apparently doomed and no-one to stop the alien force.
Later in the decade, the BBC continued its "grown-up" sci-fi tradition with Out of the Unknown. In some ways, this could be described as a British answer to The Twilight Zone, but not exactly. These were longer stories (an hour) and didn't have the quirky "house style" — this was simply a thread for sci-fi and supernatural stories, ranging from outer space to haunted houses, and from grimness to comedy, combining original TV plays with adaptations of stories by authors such as Asimov and Wyndham.
The overall standard was high, but some episodes stand out particularly in my memory. The Midas Plague, a comedy set in a future where overproduction by robots means only the privileged can work and live frugally, while the unprivileged have to meet a punishing quota of consuming goods. Immortality Inc., where life after death can be obtained, but only at a price — and the downside is that killing someone with "hereafter insurance" isn't considered murder. The Uninvited, in which an elderly couple find their flat haunted by scenes of a murder that will be committed by the next occupant. All great stuff.
Rather more sci-fi, though, was aimed principally at children. The first I remember — and what got me hooked both on sci-fi and astronomy — was the Pathfinders series (1960-61) consisting of four serials: Target Luna, Pathfinders in Space, Pathfinders to Mars and Pathfinders to Venus. These told of various expeditions, to the Moon, Mars and Venus, always with a group of children along for the ride (naturally). I'm not sure how well they'd stand up now, but at six or seven, I found them awesome.
I remember other one-off serials. The Master (1966) was a strange story which, I only discovered on researching this, was based on a book by T.H. White, author of The Once and Future King. Two children are kidnapped by a 157-year-old super-villain looking for a successor, but manage to overcome him.
Object Z (1965) told of the panic caused by the discovery of an asteroid heading for the Earth, which eventually turned out to have been faked by a group of scientists trying to shock the world into laying aside its differences. It showed the varying effects excellently, but somewhat spoilt the effect with a sequel where exactly the same thing happened — but for real, this time.
The Stranger (1964-5) was an Australian serial, long before the Australian film and TV industry came of age, about human-like aliens trying to peacefully settle on Earth without being noticed. It was notable for showing both moderates and extremists on Earth and among the aliens, and the usual group of kids have to make sure the moderates on both sides win out.
However, perhaps the second most successful sci-fi product of British TV in the 60s (after the obvious) was a succession of puppet shows. Gerry Anderson had developed his "Supermarionation" technique in shorts for very young children: The Adventures of Twizzle, about a mutant boy who could extend his arms and legs; Torchy, the Battery Boy, about an alien who fell to Earth; and Four Feather Falls, a magical western. He then turned to longer sci-fi shows for older kids. Supercar was about a futuristic vehicle, Fireball XL5 a "space patrol" story, Stingray about a submarine battling a hostile undersea civilisation, Thunderbirds about International Rescue, using their advanced vehicles to bring hope to hopeless situations, and Captain Scarlet and the Mysterons about hostile, invisible Martians. There were further shows later, but unfortunately, even by the time Captain Scarlet was on, I'd reached the "awkward stage" — too old for kids' shows and not yet old enough to realise that it didn't matter.
In general, Anderson's set pieces with the vehicles were far stronger than his handling of human characters. These moved slowly and awkwardly, though it didn't really seem so at the time. It was a shock to rewatch Thunderbirds many years later and find that the strings were visible. They weren't when I was a kid, honest. The vehicles, on the other hand, were handled with the same sort of models that would have been used in live-action shows, and they were the stars.
Still, the characters weren't irrelevant. Certain males of around my age might still get misty-eyed if you mention Marina or Lady Penelope. Or the Angels, of course. Not a bad effect for puppets.
This is just a brief survey, and I'm sure I've missed out a lot — both shows I watched and have forgotten and those I never saw. My family were never obsessive TV-watchers, and we had it off to read at least as often as having it on. And, of course, as the decade progressed we were introduced first to a strange old man in a police box, and then to a starship on a five-year mission, and sci-fi on TV was never the same again.
Still, I hope this has shown that there was a lot more going on in the 60s. Was it really the Golden Age? Probably no more so than whatever decade you grew up in, but it was my Golden Age. That's all that anyone can say.