Duel trials
Duel what now? Yes, Duel trials, you heard that right. It's not even funny how many gamers forget about this feature of the game because you barely touch it after a few sessions as a beginner. It is not because it does not offer much, it is exactly the opposite. These beginner-friendly trials offer a huge amount of good information, which can have a huge impact on someone's gameplay. It's just that you forget it, because your main objective is to fight opponents online or cultivate NPCs for goodies.
That said, it's like completely ignoring a ton of gems you might earn, right off the bat. Each section of these trials and its modes offer a handful of gems the first time you complete them. While they aren't anything spectacular independently, if you complete them all, you are sitting on thousands upon thousands of additional gems. Not only can they be vital in building your first Meta Deck, especially when you are a beginner, but they are also very easy to obtain. On top of that, more and more quizzes and trials are added over time, with a few special events on top of that that last for a limited time. Either way, Duel Trials is the perfect place to get a massive amount of gems, early on. It's also a good starting point for players who return there, after being away for a while.
Events
Explicit, Events are a stable way to earn gems quickly. Which you can do frequently, because every week the game has a different event without interruption. From bigger ones like the current Tag Duel tournament, to smaller ones like Duel Quest, there is no shortage of them. Make sure you participate in everyone, no exceptions.
PvP
For most players, this is the only reason they play Yu Gi Oh! Duel links, so it's not much of a surprise. Play against people online, earn gems, that's it. Remember that your rank resets at fixed time intervals, so you can collect more afterwards.
Improve characters
Another great way to amass tons of gems, rather quick but not really, is of course to level up your characters. There is a catch, however. It is advisable to upgrade multiple characters to a certain point, rather than focusing on just one to get the largest amounts of gems offered through higher levels. This is due to the fact that the leveling process is much faster for low levels than for the latter. By the time you could have one character at the max, you would have 4-5 (at least) characters halfway through, racking up more gems in return. Then you can definitely start to focus on maximizing them, but until then, if you want significant amounts of gems quickly, this is the way to go.
There are other ways to grab a few gems, like looking at a Replay per day which offers 5 gems each time, but since those amounts are really tiny, I'm not including them here. Just make sure you are active, enjoying all the activities in the game, and the gems will come to you in bags.