The Gold/Silver/Crystal entries were a conservative sequel to the original Red/Blue/Yellow games, recycling the original engine so that they could be turned out quickly to capitalize on Pokemania. The player is given a new continent to roam, and a new quest to become the top trainer while also seeking a mythical legendary Pokemon, but otherwise the game is extremely similar. The main new point of interest here is a fairly long list of added features, like a day-night cycle tied to the GBC's internal clock and Pokemon of different genders potentially gettin' busy at the Day Care to pop out a baby.
As with Red/Blue and then Yellow, Gold/Silver came first and Crystal came a few months later with a series of its own unique tweaks and additions. Let's talk about the new stuff common to all three versions of the game first, then in the following paragraph we'll get into the further changes that Crystal made.
All of the versions of the game now ask you to set the date when you start a new game, and the actual time of day will be reflected in the game. Fortunately, everyone in Pokemon World seems to be on a strict regimen of meth, since no one ever goes to sleep and playing really late doesn't lock anything off to you. The main thing the time cycle does is change the Pokemon that are available in the wild in some areas. That, and certain special bonus events are only available on certain days of the week. This game also introduced the "Dark" and "Steel" Pokemon classifications. Steel was added primarily as a foil to the Fighting class, and Dark was added as a counter to the Psychic class, both of which were a bit OP in the original games. In total these new games added 100 new Pokemon to the original roster of 151, with a handful unique to each game to encourage the kiddies to triple-dip, of course. The ability to swap Pokemon with Red/Blue/Yellow becomes available about 1/3 of the way into this game's story, and interestingly you can actually send them back in time to the old games so long as they existed in those games and they don't have any moves that are unique to the new games.
Whew! OK, so now here's what's up with the Crystal version exclusively. You can choose a female main character in this one if you like, and there's a pile of little aesthetic tweaks and additions to the environments. When enemy Pokemon appear in battle, they have a little animation before the fight begins. This is the only one of the three games to have the Battle Tower, and the story has actually been altered a little bit to emphasize the legendary Pokemon Suicine as a main focus, whereas he was more of a side quest type deal in Gold/Silver. The Japanese version actually has even more extensive additions than this, mostly centering around use of the then-new Pokemon Mobile System GB, which was the first introduction of online play of any form to the series. Of course, the non-Japanese territories missed out on that, but even so Crystal is probably the version of choice of the three just for the art improvements if nothing else.
Whatever game you choose, the basic story is the same. You play as a trainer who looks suspiciously like Red from the first game, but no really it's totes a different kid, and we're on the new continent of Johto to boot. The overall goal is the same, though -- eight gyms, win the badges, take on the Elite Four, be the very best that ever waaaas. Professor Elm is still lurking about and still grants you your starting Pokermang, and there's a new rival that is always appearing at annoying times and will take the guy who has the elemental advantage over the guy you picked. Unlike the arrogant but otherwise not really offensive rival from the first games, however, this one is a real stereotypical Japanime wangsty dickhole who thinks everyone is "weak" and wants to scour them from the face of the Earth using his cutesy monsters.
It's a little disappointing that there's so much in the way of re-used assets, and that you have to spend so much time in the early stages grinding on the same doofus Rattatas and Pidgeys that you were grinding on in the previous game. Not to mention all the very similar story beats -- the gym quest, Team Rocket running wild as secondary antagonists again, an also-ran from the first game now the Ultimate Pokemon Trainer in Johto for who knows what reason, and so on. But the new features really are pretty substantial, Johto is a pretty large new area, and as a bonus you get to go back to Kanto and take on their Gyms and Elite Four once you complete it. The soundtrack here is also almost completely new.
Though it's easy and definitely kiddie-focused, GSC is still a solid single-player RPG in its own right, which is the chief reason for the high rating. If you're looking to play through the series from the start, it might actually make more sense to start here and skip Red/Blue/Yellow, since chronologically you'd be running into FireRed and LeafGreen on the GBA next anyway. This story is pretty much detached from that of the first game, and this way you're not playing the same old clunky Gameboy engine twice.