Gems aren't part of any quests or special objectives in Shadow of War, but they are among the most powerful and uncomplicated power ups you'll encounter, so well worth collecting. They also have a few achievements linked to them, for those going after total completion.
There are two main ways to get gems. The first is by completing missions- some mission rewards or optional objectives will win you a gem. For the Shadows of the Past challenges, every mission has an optional objective with a gem reward.
The latter in particular is useful because these gems are already upgraded. Still, the most common way of getting gems is by killing gem-carrying orcs. They can be distinguished by the little white asterisk floating over their heads as you get close, and they'll appear as white markers on the minimap (rather than the usual orange).
Make a point of killing and looting these orcs- you'll need a ton of raw material to forge the highest quality gems, and the earlier you start, the better. You'll also want to complete the midgame Gondor quests as quickly as is convenient to gain the Prospector skill upgrade. This perk drastically reduces the number of gems you'll need to gather overall.
Lastly, while you're eyeing your minimap for white markers, watch out for loose gems as well. Sometimes you'll kill a gem carrier during a quest and not realize it, and sometimes gem carriers get killed by wandering monsters or enemy encounters. These gems linger on the map, and can be scooped straight into your pockets.
There is, unfortunately, no way to determine which gem type you get (or whether you get Mirian instead), so just go for numbers, gathering gems whenever you have opportunity, and you'll be decked out like a monarch in no time.