The very first "format" was Base Set and friends. Base Set was incredibly well designed from a casual point of view, with a lot of fun effects, fun and useful not-fully-evolved Pokémon and some of the most awesome Trainers ever. However, they obviously underestimated Trainer cards. Cards like Gust of Wind and the Energy Removal family pretty much ruined this format and turned it into a very basic-heavy format with really only 1 viable deck. Team Rocket brought a whole lot of useless cards, and the Gym sets, despite being some of the coolest sets ever, only really added Rocket's Zapdos and Erika to the mix (though Erika's Jigglypuff donk is still one of the funniest strategies ever). The Neo sets tried to fix the Trainer problem, but the removal of Energy Removal from the format and the introduction of Dark and Metal sort of ruined it in another way. I think this may have been the low point, where an overpowered Sneasel backed up by an army of abominations known as Slowking reduced the format to a single-deck format where once one player got ahead the other had no chance of coming back.
I'm a huge fan of the e-series, and one of the reasons for this is that it was the first attempt at really fixing the system. The TCG up until this point was a lot of fun at the casual level, but Trainer cards were far too good and oppressive, and there was no downside to sticking as many as you wanted into your decks. E-series added Supporters, but perhaps more importantly kept the power level in check. This sadly meant that the e-series cards never really had their chance to shine. The Neo+e-cards format was weird and not really that noteworthy. Scizor was really powerful because Metal Energies were really powerful, but other than that cards from the Neo sets (like Feraligatr+Parasect) kept dominating. The power creep following EX Ruby and Sapphire meant that the e-series cards were once again widely neglected (apart from some of the better Supporters).
EX Sandstorm brought Dunsparce, EX Hidden Legends brought Jirachi, and thus one of the first Pokémon-based engines was born. This is when the changes introduced in the e-series really started to turn the TCG into a very fun and skill-testing game. FRLG brought Pidgeot, perhaps the best support Pokémon ever. Later sets gave rise to some very fun synergies and quite a few different decks. Things like Dark Dragonite and Dark Electrode, Ludicolo and Magcargo, and so on. Decks in this era were very consistent, but not necessarily overpowered. Around this time we also got Scott, which gave us the first Supporter based engine (the Scott Engine), where 1 Scott would give you whatever you wanted, plus another Scott. At the end of 2005 we got Delta Species, which gave us an even better Supporter-based engine, the Holon Engine (focussed around Holon Transceiver). There were still a few very synergistic decks (the most notable of this era was Metagross and Dragonite), but now they didn't necessarily rely on other Pokémon for a coherent framework. I should probably mention LBS here as well, which was by far the most popular deck at Worlds that year (I know, I was there), which featured the combo of Blastoise EX, Lugia or Steelix EX, and Holon's Magneton or Electrode to do a lot of damage very fast, helped by the insanely good Pidgeot which made the deck far more consistent than it had any right to be. Decent supporters and very good support Pokémon made this an era where any deck was generally able to do what it wanted to do. There were very fast decks and more grindy decks focussed on a specific strategy, and everything in between. In 2006, LBS only won the Juniors category at Worlds that year, a weird rogue deck focussed around Manetrick EX fought a rogue deck focussed on eeveelutions in the finals of the Masters division. Heck, a Lunatune/Solrock deck won the Seniors division. While there were certainly dominant decks and dominant cards, the format felt quite fun and diverse and allowed for a lot of creativity, and decks felt like a cohesive whole rather than the sum of its parts. Solrock was perhaps the cheapest deck ever to do well (I think most lists had no rares at all), and LBS was by far the most expensive deck at the time, and both were completely viable. This is as it should be, and made this era perhaps the high point of competitive Pokémon.
The end of this 'era' was in my opinion the introduction of the no-fun-allowed cards in EX Crystal Guardians, most notably Cessation Crystal, which stopped a lot of the shenanigans of the previous sets. The message was clear: the TCG had to become simpler. This was hammered home even more in Diamond and Pearl, which like every set based on a new game reduced the complexity by a lot once more. Unlike FRLG, though, it didn't really have a lot of hidden dept (like Pidgeot), nor was it surrounded by sets which allowed for shenanigans. Great Encounters gave us a very good support Pokémon in the form of Claydol. However, Claydol was really the only "engine" a deck could have at the time, so it was pretty much ominpresent. The dominant deck of this era, the deck which also ended up winning Worlds, was Gardelade: Gardevoir + Gallade. The deck was quite skill-testing, but also very oppressive, being so obviously the best deck. I went to Worlds without an invite this year, and played the dumbest deck I have ever played in my life at the Last Chance Qualifier there (it was a Venomoth/Butterfree deck), and won my first 2 rounds, putting me in first place in the standings. It was glorious. My luck ran out, though, but I did make a grown man very angry, so there was that. This format wasn't horrible by any means, but it was a step below the previous ones.
The complexity came back in full force with the arrival of Legends Awakening, and most notably Uxie. Uxie (and a little later Sableye) once again made a lot of decks hyper-consistent, which meant a lot of weird decks could rise to the surface. This was an era where weird decks like Regigigas Lv.X, AMU (Azelf, Mesprit and Uxie lv.X) and energy-less Gyarados could live alongside more straightforward decks like Rampardos Donk, Machamp Donk, and the likes. This format felt very open. I remember Gengar being very popular in the US, while Machamp was all the rage here in Europe. I remember Gyarados being a deck here long before the US caught wind of how good it was, after which it quickly became the most popular deck there. This format didn't really offer a lot to people who love to build their own strategies (I'm in that category), but it did offer a wide array of archetypes to choose from. No matter what kind of deck you liked playing, there was a deck here for you.
Then came Platinum, and everyone started playing the SP deck. SP had a brilliantly consistent trainer-based engine, a way of stopping your opponent's strategy (and most importantly, Uxie) in the form of Power Spray, and some of the best Basic Pokémon you could hope for. So an SP deck was what you played, no questions asked (I didn't, though, but I'm weird like that). There were a few other strategies (a Beedrill-based deck won Worlds in 2009), but this format mostly suffered from the fact that the SP Engine made everything far too consistent and far too samey. Consistency is fun when it allows for a lot of different decks, not when everything has to fit within the same framework. This format wasn't bad either, but it did feel a lot less diverse. When Platinum came out, everyone and their dog played the SP deck.
HGSS tried to rid us of the SP menace in the dumbest way possible: reduce the complexity and increase the power level. This started an era of more straightforward decks and strategies. This is when I really started to lose interest in the TCG. Then came Black and White, which not only doubled down on the simplicity and straightforwardness, but also brought us overpowered Basic Pokémon-EX. And that's how we ended up where we ended up. The power level of Trainers has gone up again, but not in a way where you're building specific engines, but in a way where you just cram the best ones in your deck. Because Pokémon aren't allowed to do as much, and aren't allowed to be as complex, Pokémon-based engines are pretty much impossible and complex interactions become very unlikely. I don't play the TCG anymore, but I do follow it closely, and it seems to have lost all of the charm and dept it had back in its heyday.
Just as an aside. A couple of friends and I still very often play the TCG by shuffling a specific selection of cards (a cube) and then giving every player a pile of cards to build a deck with (kind of like at a prerelease). Throwing all the cards from different eras together proved to be horrible, since power creep meant that if most of your usable Pokémon came from, say HGSS, you almost always beat the guy who got stuck with e-series Pokémon. So I started to build cubes for specific eras. The vintage cube, using cards for Base Set up until the end of Neo, turned out to be the best cube. I was able to select a wide variety of interesting basic Pokémon (we're allowing owner's Pokémon and Dark Pokémon to evolve into their regular counterparts and vice-versa), a lot of very fun and unique evolved Pokémon and a very nice selection of Trainer cards (we've excluded Gust of Wind and the Energy Removals, but included Double Gust) which allow for some very fun and oddly consistent decks, with a lot of different speeds and strategies. It's odd how many Pokémon in this era can function both as support and as part of the main strategy. Anything from Hitmonchan/Electabuzz speedy decks to Do the Wave decks to Alakazam combo decks to even Koga's Zubat-based decks is somehow viable. This all makes me think the old format would've been very deep if it weren't for the oppressive Trainer cards of the time. The EX-era cube turned out to be nowhere near as interesting. What I think this means is that the world needs more interesting Poké-Powers and more functional not-fully-evolved Pokémon.
There you have it. I'm sure I missed a few important changes and decks, but this is what I remember and how I experienced it. Congratulations on getting through all this.