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What was science fiction like in the past? How did ancient people imagine the future?
Specifically:
Honestly, the closest thing I can think of is Horace's carmen saeculare or Vergil when he is feeling particularly mantic (Anchises speech in book VI being a famous example). Both foretell visions of a future golden age of peace and prosperity and good morals and the like. However, both are also clearly addressed to Augustus, and more importantly play off of very old tropes of praise literature, with echoes as far back as Mesopotamia. So we can't say these really show authentic Roman predictions of the future, but they do show how Romans could use visions of a utopian future, and what that future would look like.
Of course, you also have Ammianus Marcellinus, who concludes his work by, essentially, saying that things are in the toilet and the handle just got pulled down.
It will depend on the Roman (or Greek). As /u/Tiako writes, some Augustan poets characterized the future as a golden age, mirroring the mythical golden age of the past, though this vision is deeply tied to Augustus' propaganda that he was "restoring" the Republic, religion and morals after decades of civil war.
More interesting for you, maybe, would be how certain political thinkers thought about the development and future of mankind. Two of them come to my mind from the Roman period. One is the second century B.C. historian Polybius, an Achaean Greek who lived at Rome for much of his life and who wrote a world-wide history of the rise of Roman power down to the year 146 B.C. The second is the first century B.C. poet Lucretius, an Epicurean who wrote a six book poem titled "On the Nature of the Universe."
Book 6 of Polybius' histories is about the Roman politeia ("constitution", though it's not a perfect translation) and why it is superiour to those of other states. In writing this book he presents a cyclical vision constitutional change and of human history:
Polybius 6.5: http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Polybius/6*.html : "What then are the beginnings I speak of and what is the first origin of political societies? When owing to floods, famines, failure of crops or other such causes there occurs such a destruction of the human race as tradition tells us has more than once happened, and as we must believe will often happen again, all arts and crafts perishing at the same time, then in the course of time, when springing from the survivors as from seeds men have again increased in numbers and just like other animals form herds — it being a matter of course that they too should herd together with those of their kind owing to their natural weakness — it is a necessary consequence that the man who excels in bodily strength and in courage will lead and rule over the rest..."
The key phrase for me there is "and as we must believe will often happen again." So just as for Polybius, kingships will turn to tyrannies which will turn to aristocracies which will turn to etc..., and just as every state has a cycle of strength to weakness, so too does the whole human race have a cycle of reverting to an almost "state of nature" due to natural disasters and then to rebuilding political society again.
Lucretius in book five of his poem narrates the creation of the world and the development of animals and mankind (all in a way that precludes divine intervention: Epicureans did not believe in the gods and thought that the world was made up of only atoms and void). A link to a translation here, though unfortunately there are no line numbers. Ctl+F "But the race of man then" to get to the history of mankind part: http://newepicurean.com/suggested-reading/lucretius-de-rerum-natura/munros-lucretius-book-v/
Lucretius describes the development of mankind from nomadic roots to the rise of civilization and societal institutions, and tackles a lot of technological progress along the way, such as the use of fire and language. Lucretius argues that nature and reason prompt men to adapt and pick up new technologies and new ideas, including justice, art and religion. He ends the book with:
"Thus time by degrees brings each several thing forth before men's eyes and reason raises it up into the borders of light; for things must be brought to light one after the other and in due order in the different arts until these have reached their highest point of development."
Lucretius doesn't deal with the future per se, but as far as I know this is one of the few ancient texts that deals so directly with technological change. That's interesting for us moderns, because technological change is key to how we view the future due to the rapid advancement of technology in our own lifetimes. It, however, did not generally play a part in Roman conceptions of the future, which (they assumed) would look pretty similar to the present. But I think you can make a case that Lucretius at least, even if he talks about the "highest points of development", was able to conceptualize a future in which nature would prompt mankind to adopt new technologies and concepts.
This is all probably a much longer answer than you wanted, but I think both Polybius and possibly Lucretius hint at very different but interesting visions of what the future of mankind would have in store, especially when you compare Polybius' cyclical pessimism and Lucretius' natural progression with the optimism of Augustan age poetry. And I'm sure that there are other Greek and Roman writers who had very different but equally interesting conceptions of how the future would turn out.