Civ 4 marks the end of the 'old' Civilization games, and honestly I don't think there's much reason to look further back than that if you're just joining the series. It's fully-featured, cheap with all expansions, and offers a great experience.
Civ 5 was the start of the new era, but has been out long enough to get two much-needed expansions. It cleaned up the look and interface of the game substantially, and changed the feel of combat to a more modern TBS approach. I think common consensus is that it's a bit less deep than 4, but plays more smoothly and with more engaging combat.
They're both great, and I don't know that it matters too much which way you go. If anything about either game frustrates you, there will be a mod to change it. You really can't go wrong.
Civ 6 is more controversial - it reworked a lot of the city-building elements really substantially, and I'm not sure it's for the better. It'll probably be a solid game by the time all the expansions are out, but right now it doesn't feel complete and polished the way 4 & 5 do.
Don't get Beyond Earth. Honestly, it's just not worth it - it doesn't have any real merits beyond 5 except "in space!", and it's long on balance and design issues.
But I still hugely prefer it to the doomstack approach - wars actually involve doing something now.
Of the mainline series, 1-4 is a straightforward progression that doesn't rock the boat too much. Civ1 is still fairly easy to get into but has a lot of crude, unbalanced or easily exploitable elements(e.g. the AI has a random chance of building a Wonder of the World every turn instead of having to actually allocate a city's production to the project over many turns like the player). Civ2 has fewer exploits and more polished 90's-era graphics, but now the number of units and cities explodes into a very tedious, micromanagement heavy late game - the degenerate strategy of Civ2 is called "smallpox" because it involves spamming a lot of small cities across the map. Civ3 added a lot of additional balancing and finetuning and graphical flair, and Civ4 does more-or-less the same. I have trouble remembering Civ3 after Civ4, really. But the basic feeling remains consistent: Most Civ strategy is focused around optimizing fine details across your entire empire towards a focused direction, so that you survive now and gradually get ahead of the AI players later. And the late game slows down because a lot more is going on in each turn. It gets hard to keep track of things.
The games after that make more drastic changes. They still feel like Civilization, but with major changes to the cities, board and combat components that differ from the "Empire" model that had been employed since the first game. The ways in which you have to make tradeoffs of "now" versus "later" or the kinds of capabilities to emphasize are more varied. And they have higher system requirements - Civ5 at release was quite heavyweight and buggy, and Civ6 has gone similarly. But these are things that everyone expects will get smoothed out with time. With the newer Civ games, it's more a case of the designs just being different versus being definitely better, with emphasis placed on different things.
Some vocal long-time users of things tend to dislike newer versions of that thing, no matter what it is. Personally I started with Civ 3 until I tried Civ 4, and then it just felt outdated. I liked Civ 4 until I tried Civ 5, and then it felt outdated. I liked Civ 5 until I tried Civ 6, and then it felt outdated.
If you started with Civ 5 or Civ 6, I don't think you'd be missing out or disappointed with anything.
It has the particularity that it has an optional "extended game" where the tech tree goes further into the future. Instead of stopping developing technologies at the present time, you could develop cyborgs, plasma weapons, force shields, and stuff like that. And you could go to Alpha Centauri, where you would find a different map and an alien civilization there. IMHO no game in the series has done the future so well.
As a bonus, Test of Time lets you play the original Civ2 game with improved graphics if you don't like the extension, and it also includes space and fantasy scenarios that are not bad at all.
Why almost no one knows that game? Probably the reason is that it was panned when it came out as it was widely seen as a cash grab. It came out a few years after the regular Civ 2, and it didn't feel good that they were charging the price of a full game for what was essentially an extended remake of the game people had spent their money on previously. But now that we are in 2018, who cares about that. In my view it's the best game in absolute terms.
Civ 3 introduced many new concepts, but some of them didn't worked too well and the playability was not too polished IMO (culture, and absurd lack of basic resources like iron and coal, I'm looking at you). Civ 4 continued the trend of Civ 3 but polished it, nerfing or buffing mechanics that were unbalanced, and it is the game I would rank as the second best of the series if you don't want to play Test of Time.
Civ 5 made some quite radical changes mentioned by other commenters, while it's still great I personally don't find it as fun as the earlier games. And the same goes for 6.
Also, RIP Civ World. Not quite a Civ game, but maybe the best Facebook game I've ever seen. It would have been great if they had bothered to solve the bugs. Some of us still remember you.
In particular I found it way too easy to get a tech advantage and march across the map in 5.
Civ 4 is the best version of "classic civ", so if you want a modern take on 1, 2, or 3, go with Civ 4.
Civ 5 was dramatically re-developed. Instead of a square grid (4 possible movement directions for a unit) they moved to a hex grid (6 directions!). And they removed unit stacking, so you don't have these army hoards on single tiles that can wipe out everything in their path. They changed a lot and I think it was for the better - a lot of quality of life improvements. It plays much better than 4 while still retaining the classic civ "just one more turn" feeling.
Civ 6 is too much. It's got too many features and too much user interface. It feels like a chore to play some times. The music is better in Civ 6 than 5, but nothing beats Civ 4's Baba Yetu.
That's true in the same sense that it's true thst I did all the vocals on Civ 1.
Civ5 brought some significant changes to the gameplay but you are in the great position to make an unbiased choice. I loathe it (playing since the first release does that to you) but I have friends who prefer the new gameplay.
By the time Civ 5 came out in 2010, most series fans would have been playing fully-expanded Civ 4, with all the rebalancing and new content that entails. Anyone who liked vassal states, Great Persons, corporations, or the complex space and culture victories was going off expansion content.
First-release Civ 5 made some interesting alterations to the series, but had a lot of implementation issues. Unbalanced wonders, AI bugginess, and a late-game slog often made it feel bad regardless of the design choices. BNW-era Civ 5 brought a lot more logic to the tech tree, smoothed out broken features like "one free technology", and made the late game vastly more interesting.
Honestly, I suspect we've only recently gotten to a 'fair' comparison of 4 and 5, with both of them feeling like classics in light of BE and 6.
Let's just pretend BE and 6 never happened.
I'm not sure how else you explain screwups like "will trade 9 gold now for 1 gold-per-turn over 10 turns". It's like they understood the concept of a loan but got the system backwards.
Or military strength judgements, which are blatantly just running on "number of currently active units". You can be feared for having lots of spearmen in the Industrial Era, but the ability to crank out 10 high-end infantry per turn isn't even present in the assessment.
It's a bizarre and embarrassing weakness for a series most people play in single player. These aren't grand strategic errors, they're "make the number bigger/smaller" errors that random modders handle vastly better than 2K.
It's funny seeing Civ V so praised, because it was a decisive game when it was released. The AI bugs you mentioned were occasionally annoying, but overall I loved V - the combat changes alone made the game better then IV, but I always felt like a bit of an outcast when people would rag on the game.
I feel similarly about VI now. I really enjoy the depth added to the city changes and how they cleaned up the UI. I hope it gets a similar hindsight appreciation as V.
There was so much polish in other areas that it was odd that the rough edges were never sanded down. The bad AI & especially diplomacy were one of the most immersion-breaking experiences and unlike some of the other tweaks they haven’t been well solved by community modders.
5 is a perfected 4. 4 Will forever be the GOAT because of Leonard Nimoy's narration, but 5 is technically the better game IMO.
I also liked IV and I and all the newer ones too (I never played III because IV came out before I got my hands on a Mac port of III). The main negative point of the newer versions is how slow turns get toward the end on large maps with lots of civilizations.
That's not exactly new; it's just if you're playing Civ I or II today, turn processing takes maybe a hundredth of the time it did at the time of release.
I need a new computer :D Or maybe just play Civ IV instead.
I literally did not study for history from grades 6-10 because I had learned 80% of the content by reading Civilopedia entries during gameplay. It's been a huge inspiration later in life as I've thought about educational gaming -- both as a classroom teacher and now as a technologist.
PS - I still quote CEO Nwabudike Morgan and Chairman Sheng-ji Yang on a monthly basis in adult life
I think I'd probably recommend Civ4 or Civ5 as a starting point. Civ4 is the "old school" style, Civ5 is a very boardgamey experience. Colonization, modulo the historical issues I described in my other post, could be a good fit for the more economically-minded.
Make sure you use the latest community AI mod since the original AI is usually not great (mediocre in Civ4, bad in Civ5 and nonfunctional in Civ6).
If you get stuck refer to this: https://lparchive.org/Civilization-2/
As someone who's been playing since 3, I consider those people tragic victims of nostalgia. :P The newest game, 6, is great and strictly better than 5 (I could make this a very, very long comment explaining why, but I'll spare you), though with one exception: the mod scene on 5 is much more mature, so if you were considering modding the hell out of 5 you could make something that rivals 6. In contrast, it's hard to compare 6 to 4, because they're really such different games (5 was sort of a conceptual break in the series). If you don't mind the dated graphics then 4 is super solid, and I think you can get it on Steam for $20.