OH OH ME PICK ME! I CAN ANSWER!
I'm completing my Master's thesis on just this topic. I've been waiting for a moment to answer so I can procrastinate.
So, first of all a good deal of my info will come from Kyle Harper's The Fate of Rome an excellent read on the collapse of the Western Empire and how climate was a major part of that collapse.
What we can tell from the 1st and 2nd centuries is that the life of the average Roman, whether in Iberia, Gaul, Italy, Palestine, or Egypt, seemed to have had a fairly decent life (comparatively of course).^(1)
For the most part life was good. He even points out that the Western Empire saw the construction of towns "*ex nihilo"*^(2) Life was good. A lot of this seems to be due to what some Paleoclimatologists have called the "Roman Climate Optimum" which is precisely what it sounds like. There have been plenty of studies done on tree rings, lake levels, and ice core samples that have corroborated the written sources we have from this period describing the abundance of the period. For some basic info check out this article ("9,400 years of cosmic radiation and solar activity from ice cores and tree rings" Steinhilber et al)
But, you asked about the Crisis of the Third Century. And I will deliver!
The climate seems to have dipped a slight bit; which had a nearly cataclysmic effect on the Romans.
By the mid-third century the Nile RIver seemed to refuse to give its regular flooding events and dried up on multiple occasions.^(3) It has been said on multiple occasions that Egypt was the breadbasket of the Empire, and with that vital source of water cut off that would keep the people fed widespread troubles began to unfold. Drought struck throughout Palestine and multiple other regions of the empire.^(4) People, understandably would be shaken by this, but that wouldn't be the end of the tremors.
Plague ripped the empire in a way it had not experienced prior, and unfortunately for them, would not be the last. The Plague of Cyprian devastated the people.
All of this taken together it's no wonder that Cyprian, a Bishop from Carthage, remarked that the world was old and faltering. "The falling rays of the setting sun are not so bright or brilliantly fiery" and further on "The fountain that once overflowed from abundant springs , now forsaken b age scarcely yields a drop."
Not only that, but in Palestine (even in the Talmud itself) we see increasing references to the lack of rain and the growing intensity of droughts within the land and the assumption that there was some sort of DIvine recompense due to the people for the their transgressions of the Law.^(6) In the Sperber article he does an excellent job of cataloguing these episodes in chronological order, it's a good read if you're interested in Talmudic studies.
Factor in the Persian invasions and incursions made by the Gauls into Roman territory and you have a recipe for, well, a crisis and the idea that as the world was growing old, then Rome itself perhaps would not be as eternal as thought.. From what I can tell most people were very well aware of the problems facing the world, but they didn't seem to view from a geopolitical/environmental perspective, and of course how could they, especially with the second one?I realize this is brief, it's a small part of my Master's Thesis (mostly focusing on synagogues featuring Zodiacs built in the 3rd century) that's why my research is so slanted towards Palestine, but I hope that it somewhat informative and helpful!
/u/harristotle27 has told you quite a bit about the environment and what people would experience during the third century: I'd like to take a closer look at what they said and wrote about their experiences and what they expected to happen, with a bit of historiography thrown in for good measure, because you've actually stumbled upon something of a debate.
So! There is a very famous article by Alfoldi from the 1970s, 'The Crisis of the Third Century as Seen by Contemporaries', which I would urge you to read. It is a very interesting text: Alfoldi surveys writing of contemporaries from a relatively broad basis of evidence, drawing on peasant's petitions as well as Christian polemics, to argue that there was in fact something like a crisis in the third century, and that people were very much aware of it. Some of the highlights include a petition of peasants from Asia Minor under Philip the Arab, where they refer to their past as 'an age of necessities and upheavals'; Herodian, a civil servant of Greece, whose 'conception of the period of Roman history after the death of Marcus Aurelius was a continual series of heavy catastrophes, interrupted at best by short breathing-spells which were followed, however, by worse periods, with the danger that the Empire might collapse': and two Christian authors, Origen and Commodianus, the former convinced that the empire has a future, while the later, ten years later, expects the end of the empire in near future because of barbarian invasions, military revolts and persecution of the church.
"Wow, mpixieg, that sounds like precisely what I asked about, why should I read anything you will write when I can refer to this very competent scholar instead?" Well, the problem with the article is that even though Alfoldi argues that people of the third century have reached the conclusion that they were living through a crisis of the Roman world, other people have disagreed with him on that. Karl Strobel in 1990s takes an extremely critical view of the evidence provided by Alfoldi, arguing that what this actually shows us are the immediate reactions of contemporaries to particular events, and not a sustained view of their present as being an age of 'crisis'. He'd say that the peasant petition only shows you that recently they have experienced some problems (and we know that they are actually complaining about the passage of soldiers who confiscated their property, so perhaps it is referring to that): and actually, when you think about it, the fact that they are writing to the emperor shows that they have quite a bit of confidence in the empire - you don't petition the emperor if you don't think he can solve your problems. Similarly, he'd point out that the trope of the current period being an age of decline is absolutely something that Roman authors like to at all times: whether you ask Sallust or Tacitus, they'd be keen to tell you that their age was absolutely awful and worse than the previous period. Finally, in regards to the Christian texts, he says that they are more of an answer to the Decian persecution than to any sort of 'crisis of the third century': this is why they have this rather apocalyptic tone on occasions. Fundamentally, what Strobel believes is that one can only diagnose a crisis when it is over, and that the Romans - while recognizing that bad things happened - did not assign them to a longterm diagnosis of their empire as in a period of sustained crisis.
So two questions there: what did the contemporaries think about the future of the empire and did a crisis happen in the third century (because what Strobel is ultimately getting at is a suggestion that 'the crisis of the third century' is a historical construct). Addressing the latter question first, I think it's fair to say that the contemporary scholarly consensus is that something like a crisis did happen, and that the empire that emerged out of it was significantly different than the one we started out with. Although there are some questions whether the term 'crisis' is appropriate (because, throughout historiography all the way to Gibbon, it has implied a value judgment that says that the later Roman empire is just not as good as the principate of the first and second century) and some people would prefer the term 'transformation', very few people would actually take the position that there were not sustained issues plaguing the empire in the period from Alexander Severus to Constantine. Wolf Liebeschuetz has a good article summarising this debate, called 'Was there a crisis of the third century?': it should come up if you google it.
Second, whether you'll go with Alfoldi or Strobel and say that people were either aware or not of a longterm sustained crisis (and I think both have quite a few good points, but what Strobel gets very right is that you have to be extremely attentive to the context in which they say these things), both can get you to the idea that either way, people were quite often actually rather optimistic about the future of the empire. What always strikes me as extremely interesting is the faith that problems can be overcome, if the empire falls into hands of a proper, good Roman: going back to the petition I mentioned, Philip the Arab rules right in the middle of the crisis, but what those peasants actually say is that 'in your most blessed times, most pious and faultless of emperors, everybody leads and undisturbed and tranquil life, because all kinds of wickedness and harassment has been brought to an end.' Is it conventional flattery? Sure, but it also shows you that these people think Philip can do something about their plight, otherwise they would not write to him. Even if parts of the empire secede, as the Gallic empire does, for instance, no one is rejecting the imperial framework: Postumus issues extremely conventional Roman coinage, with the description of REST(ITVTOR) ORBIS. He wants to be a Roman emperor, not anything else, and this is quite striking in contrast to the fifth century, when people are perfectly willing and actually prefer to become kings instead. It's important not to push this too far: some people, as we have seen, were extremely pessimistic, but this is not necessarily the whole picture.
To sum it up, if you are a Roman citizen in the third century, what you think about the empire probably depends a lot on where you are living, who you are and what year it is: senators like Cassius Dio would be influenced by their personal successes and failures in office, while the ordinary denizen of North Africa in 250s would be doing a lot better than the people at the Rhine frontier, and this would also influence what they thought. Due to this, you get a whole range of opinion from the post-Decian Christian eschatology to optimism of the imperial panegyrists: and it is important to take into consideration both, while keeping an eye on how historiographical opinion has influenced our interpretation of these sources.
For further reading, I'd recommend: A. Cameron, The Later Roman Empire (1993), John Drinkwater, “Maximinus to Diocletian and the ‘Crisis’” in The Cambridge Ancient History, vol 12 (2005) - the whole volume is very good - and L. de Blois, ‘The Third-Century Crisis and the Greek Elite in the Roman Empire’, Historia 33 (1984). In addition, consult the works I referenced above, namely Geza Alfoldi, “The Crisis of the Third Century as Seen by Contemporaries,” GRBS 15:89-111 (1974) and Wolf Liebeschuetz, 'Was there a crisis of the third century?' in Crises and the Roman Empire (2007).