Undoubtedly, ever since the first hominids there have been interactions with the felids. Most likely these would probably have been lions, leopards and probably some extinct species chasing our ancestors. However, it is possible this wasn't just early hominins getting munched by the big cats in Africa. A modern tribe in Africa (the Dorobo), are known to go and chase lions from their kills. There is obviously lots of room for debate on how successful our smaller ancestors would have been at this against predators that haven't been under pressure from humans for many millenia, but there is no reason to believe it's not a viable option.
All of this is lovely and speculative, and its not until about 30-32,000 years ago that the first evidence of humans appreciating the cats in their environment appears. The cave paintings in the decorated cave of Pont d'Arc (known as Grotte Chauvet-Pont d'Arc, Ardèche), in South France show many images of cats, particularly those of of lions (probably cave lions: Panthera leo spelaea) and one described in all the images and text I could find as a panther (which as a species doesn't actually exist, so is probably a leopard).
With the domestication of the cat beginning, its perhaps unsurprising that they soon entered into mythical status too. Egypt was amongst the first places to raise cats to god status. Cats in Egypt were known in ancient times as Mau (I can't find out if that's because of the meow noise they make though, although there is now a breed known as an Egyptian Mau), and the first god feline (a lion) was Mafdet. Another called Bastet started off lioness-like, but through time became associated with domestic cats. It was this goddess that lead to the mummification of cats that were included in human burials. Indeed the Egyptians worshipped cats to such a level that when one died it was mourned much as the loss of a human relative within the household. However, from 390A.D. when their worship was banned, cats fell from such revered status within Egypt but are still tolerated to present, and remain admired due to their significance in Islam (particularly for their cleanliness).
The lion appears over and over again as a symbol of gods, and thus equally royalty (who were god kings/queens). This is probably due to their large size, fearsome reputation and historical presence across all of the regions where early civilisations emerged. Think back to Hercules slaying the Nemean lion (which was in Greece). Thus statues of lions appear guarding entrance ways to cities and palaces across the ancient world.
The Greek and Romans are probably the most responsible for bringing most of the felids of the Old World to the knowledge of the general public. The Greeks with the expansion of various empires across the world, and the Romans followed suit. The Romans went a step further by bringing animals from across the empire for various spectacles and games in the arenas (whether small amphitheatres, or the Coliseum). Because of the array of them known to the Roman world, they appear throughout their culture, whether in statues or in mosaics, or even in sarcophagi.
King John who would reign following Richard I death, would officially set up the royal menagerie which would go on to hold all manner of felids, and open to the public in the first zoo (of sorts). 600 years later it would be closed by the Duke of Wellington, and the animals distributed between Dublin and London Zoo. Until this day the zoo maintains several felid species, although most individuals are now located outside of the city in the larger Whipsnade zoo. Interestingly, there is currently a break from tradition in London, with the Asiatic lions being relocated temporarily whilst their enclosure is renovated and thus there are no lions being present in London for the first time in 800 years. The lions remain very much an English thing (although sometimes they were called leopards historically on the seals/heraldry), so much so that the Three Lions remains the nick-name of the football team, and the rugby team combined from the UK and Ireland is known as the British and Irish Lions.
Having focussed very much on the UK and Europe so far, its time to spread further afield. Whilst the lion has been the symbol of European royalty for millennia, it also has represented royalty in Africa, with Ethiopia probably the best example. The lion featured until 1975 on their national flag as a symbol retained from their days as an Empire. This lion is much more significant as it is the Lion of Judah and can be traced back as a symbol of the tribe of Isrealites of Judah, descended from the 4th son of Jacob in the Torah. The royal lineage was traced (supposedly) all the way from Solomon to the leaders of Ethiopia until Haile Selassie. Due to Haile Selassie being important in Rastafarianism, the lion is an important symbol to that religion too. The Lion of Judah has also been added to the official emblem of Jurasalem in 1950 with the foundation of it (or at least the western bits officially) as the capital of Israel. Whilst jumping around, but mainly in Africa, there are some interesting tribal traditions that remain. The most famous is probably the Maasai tradition where boys were supposed to kill a lion as a show of their warrior strength before becoming a man. This tradition has mainly died out as it illegal to kill lions, but there remains great prestige for those who do kill a lion in defence of livestock.
In the Americas, the jaguar (as the biggest cat there) has featured in all MesoAmerican cultures. In the Olmec culture, jaguars appeared regularly as anthropomorphised, and possibly even werejaguar like. The Maya adopted the jaguar as various gods, and as with the lion in Europe, became associated with the ruling class who often wore their pelts to distinguish them, and the ruling families often incorporated the jaguar into their names. Indeed the Maya even developed a stringed musical instrument that when played makes a noise identical to a jaguar growl. The Aztecs followed the tradition of revering the jaguar, with the elite class of soldiers being Jaguar Warriors. To attain the status a soldier had to capture 12 soldiers in 2 consecutive battles (deemed more honorable than killing).
So that roughly covers much of the older history of cats that I found. Undoubtedly I will have missed some big details and many views on cats. Whether its black cats and their bad luck, or covering humans hunting cats, and cats hunting humans (e.g. lions of Tsavo), or indeed humans using cheetahs to hunt. I feel my views on that may be very biased against humans in that respect. After all, many of the lion subspecies are endangered (or extinct); cheetahs inhabit a tiny fraction of their former habitat; leopards have had regional extinctions; tigers have had 3 subspecies go extinct and several other subspecies critically endangered (there are more tigers in Texas than there are in India!); pumas are much reduced across a lot of its former range; and the story is the same for jaguars too. And that is just the big cats, many of the smaller ones are experiencing much the same situation.