People have been discovering fossils for about as long as there have been people. Curious bones appear in mines, or in hillsides after storms. You can climb a mountain and find bewildering layers of seashells hidden in the rocks. You can walk through the trackless snow and stumble upon a frozen hairy elephant. You can find eggs turned to stone in the middle of the desert.

All of these are awe-inspiring moments even with explanations ready to hand today. Imagine the profound effect this would have on people who spend their whole lives attuned to the natural world, trained since their earliest days to understand its rhythms and rules.

We cannot say anything for certain, because we have no concrete proof, but there are delightful hints of this wonder through the legends and folklore of cultures across the world. Were tales of giants taken from the interpretation of mastodon bones? Were dinosaur skulls presented to emperors who then ground them up to drink in dragon elixirs? Are Indonesian tales of Ebu Gogo, or European tales of trolls and ogres, taken from the discovery of hominid skulls? We have some traces of curios and magical items, and there's a real argument to be made that these artifacts inspired deep thought about a prehuman era. Did they inspire real scientific inquiry in our preliterate ancestors? No one can responsibly say "Absolutely yes," but if you want to explore the circumstantial evidence take a look at Adrienne Mayor's The First Fossil Hunters.

To date a dinosaur in the way you mean, we have to first establish when people began thinking about the distant past as truly distant, in a way completely alien to human experience. Most of the credit should go to the Scottish scientist James Hutton, who basically developed geology as a discipline. In the 1780s and 1790s, his theories of rock formation and observations of ongoing phenomena led him to conclude that the earth was untold millions of years old. It took a generation for his ideas to really take hold, thanks largely to the efforts of Charles Lyell. (For an in-depth discussion, I would recommend James Repchek's book The Man Who Found Time.)

By the time of Lyell, however, many scientists were working under the assumption that Earth was millions of years old. The Rev. William Buckland, for example, won plaudits for his arguments in the early 19th century that the Book of Genesis could be read figuratively, that Creation lasted for a vast span of time before the records of the Bible, and that of course while Noah's Flood was a real thing, perhaps it was not a global phenomenon (since he found no evidence of it in a cave showing thousands of years of animal occupation).

By this time, coal mining and quarrying had industrialized excavation, and the occasional uncovering of an oddity had become a regular occurrence. Buckland, as the foremost British geologist of his time, examined many of these specimens.

In 1824, Buckland revealed his reconstruction of the dinosaur Megalosaurus, the first described dinosaur species. Charles Mantell revealed Iguanodon the next year. Soon, the world was abuzz with news of similar discoveries. The flood of evidence soon made it almost undeniable that the world was astoundingly ancient.

As to how the dates of previous geological periods were narrowed down and defined, and how paleontologists and geologists worked to date their fossils and the stratigraphic layers in which they were found, I will have to pass the baton to a specialist.