Missions are one of the best ways to earn lots of Festival Coins. Once your Festival Plaza is at least Rank 4 you can do Missions. Talk to the Dancer by the bridge and you will receive three Festival Tickets daily. Use these to host missions.
Joining missions will not cost you a ticket. When you're connected to the Internet, occasionally a red banner will scroll across your touch screen informing you of a mission being held by a guest on your guest list. Tap to join and rake in the FC! Since it doesn't consume Festival Tickets, join whenever you can.
There are hundreds of different missions, but they can be lumped into a few categories. Within each category all the individual missions will operate the same.
Certain missions require you to participate in at least 20 Missions beforehand so if you're not seeing some, just keep on doing Missions and they'll pop up eventually. Don't give up!
Language Lesson: Run by your visitors and look at the speech bubbles. Talk to those speaking in the language targeted by the mission. For example, target "Hello!" and "Thank You!" for English Lesson.
Find [x]!: This can range from "Find the oldest sibling!" to "Find Bug-type enthusiasts!" Run by the visitors and look for speech bubbles linking to them fitting the status requirement and then talk to them to earn points. For example, Bug-type enthusiasts will be talking about various Alolan Bug-type Pokemon.
Research [x]!: You will have to research clothing colors! Talk to trainers wearing the specific color targeted by the mission.
Send people off to... : You will have to send visitors to various facilities and try to make them like it!
Costumes and Disguises: Find and talk to visitors wearing (silly) costumes or disguises, or as an inversion, talk to visitors who are not disguised.
Life and the Universe: A general lump category of missions that usually assign you to finding out about people's lives. For example, you might be tasked with talking to people who have free time and find out what they do and how much they have.
Type Match-Up Tests: There are two missions, that are inversions of each other. Both involve talking to all the visitors in your Plaza and answering their question which involves knowing the type chart like the back of your hand.
The first mission will involve answering using the standard type chart, the second mission involves answering the questions as if inverse battle is in effect (i.e. all super effective / ineffective matchups are flipped, so Water is super effective against Dragon, for example).
If you are looking to maximize FC gain within a short period of time, definitely run the Type Match-Up Tests. They are easy to do and plenty of other people will be hosting them too, giving maximum chance for lots of FC. The other missions will often rake in far less points for your time, and thus give less FC. It can be fun, though.
At the end of each mission, you'll be scored individually and overall as a group . Afterwards, you'll be given an amount of Festival Coins corresponding to your individual and total player score.
If you got 5 stars for both, you'll get the highest advertised FC; otherwise, you'll get a portion of the amount. In general, you should focus on your individual performance and aim for 5 stars. That way you'll always stand to get half of the maximum amount of FC, which is usually a decent number. And for extra brownie points you might show up in the Top 3!
This is because the total player score needs to be really good to get close to 5 stars--and the only way for this to happen is if there are a lot of players. However, the sad truth is that not all missions get filled up with players. Settle for what you get and try your individual best!
Global Missions are technically the exact same as normal Missions, but they are released by the head honchos of the game themselves: Game Freak and Nintendo. They will last for about a year after the launch of a game, after which time they will cease to exist.
They are released through Global Link and Game Sync so you will need to register a Game Sync ID through the Global Link website to participate. Released infrequently, Global Missions last a certain duration, after which time you will receive FC as a reward.
Succeeding in the mission will reward you with up to 4,000 FC if you are registered to Game Sync. Be sure to sign up for them every time they come out. Even if the goal isn't met you will still be gifted over 1,000 FC just for trying and being registered!