Events occur regularly in Shelter 69 and many include a new side story, H-scenes and other rewards. Events do not repeat, but old event stories and artwork can be unlocked from the Album using Vault Coins from Vault Opening events.
So far there have been four different types of event in Shelter 69, though they all have similarities. Each contains three stages that require completing a number of tasks - such as, "Spin the Roulette X Times", "Kill X Enemies", "Enhance Item Rarity X Times". For story events, completion of each stage unlocks the next portion of the story along with an H-Scene. Other various rewards are available during the event as well.
Picking Sides!: In these events you can choose which stage you start with, usually picking from two stages. Once you complete both stages, the third stage is available to play. Along with the H-scenes at the end of each stage, various awards are earned after completing tasks during the event.
Shelter Tales: These events are more straight forward since the stages must be completed in order. However, at various points during each stage you may choose between two tasks, with the second task allowing you to "skip" past an additional task. As you complete tasks you unlock additional rewards, along with unlocking the next part of the story at the end of each stage.
Hunting Party: In these events you are given a set of five missions to complete - selected from the campaign missions you have unlocked. The available missions refresh every 8 hours. You can complete the missions with Autofarm.
Vault Opening: These events are more simple than the story events. Each day you are given a new set of tasks to award event points to unlock prizes. Along with including a new H-Scene they also provide Vault Coins to unlock old event content and other special content.
Be Prepared
Try to keep 700-1000 nanites saved up so that you have plenty available during events. You will almost certainly buy Epic Crates and need to do extra Roulette Spins.
Save Daily Quests
It is frequently the case that one of the first tasks in the event will require completing a full day's worth of Daily Quests. If you collect the rewards from those quests before reaching this task you will be stuck waiting until the next day to make any further event progress. Wait until you need them for the event, or you're done playing for the day before collecting the Daily Quest rewards.
Check the Wiki or Discord
The tasks required to complete an event are posted here and you can ask for help in Discord, too. This can help you plan ahead - such as knowing to wait to buy Epic Crates or starting Expeditions or Item Enhancements early.
Start Tasks Early
If you are using Discord or the wiki, you'll know when tasks like "Level Up a Character", "Enhance Item Rarity", or "Complete Long Expeditions" are coming up. You can start these ahead of time and then wait to complete them once you are on that task.
Save Crates
Don't buy your daily crates from the shop until you are sure you won't have any more tasks that day that require gathering Trust or opening crates.
Use Champion Skills
When you need to use Champion Skills for a task, make sure to take Jane. Her skill refreshes at the start of every fight, allowing you to get at least one skill per fight. Try to pair her up with your weakest Champions that can survive the mission, possibly giving them weak weapons. This will help make the fights last longer, allowing for the additional Champions to use their skills during the mission.
If you have strong Champions or tons of eLDi, a faster option is to send a team with no weapons and skills that do no damage. Andrea, Martha, Karin, Kim, Tess, and Beatrix are all good options, but any Champion that does no damage with their skill will work. Fights will take a long time (that's the whole idea), but you can potentially complete the task in a single mission.
Kill Enemies
If there's a mission you're already farming for nanites, gear, crafting items, or resources - keep farming it. But if you just care about a lot of enemies, Campaign Missions 1-7 and 1-8 have a lot of fights against weak groups, take very little Intel and you probably unlocked them on the first few days of playing. However, if those missions are unavailable, for reasons, Campaign Mission 2-7 has 13 enemies and is a decent alternative.
Spin the Roulette Wheel
Avoid these if at all possible. If you can't avoid them, there's not much you can do other than check in as close to every three hours and possibly consider spinning an extra time for 25 nanites (but that gets pricey).
Survive Invasions
Much like spinning the roulette, avoid these if possible since there is a four hour delay between invasions. Also, if you know a task is coming up (check the wiki!) that requires invasions, avoid collecting from the radio until you reach that task. It still might occur randomly, but collecting from the radio will absolutely trigger it.
Click Activity Bubbles
Only Champions assigned to work in Production Rooms get Activity Bubbles, not Workers. Make sure all your Champions are working... and then watch and wait.
Level Up Champions
Despite what the task description says, leveling up any character works. So if all your Champions are maxed level, or you just want to complete the task quickly, you can always invite a new Worker, level them up to level 5, kick them out of the Shelter, and repeat. This is also a good way to complete Event Tasks requiring "Skip X Times". Skipping the wait time on low levels only costs 5-10 nanites.
Equipment Tasks
Remember - "Enhance" means to increase the rarity of an item, "Upgrade" means to increase the level of an item. There are tasks for both.
Workers Needed
Did you received an worker from Radio Station/Achievement? Don't send her off yet!
Rare (Blue) and Epic (Purple) workers are sometimes needed during events. You will likely need to invite them to complete some tasks. So, always keep some by the gate, then when it's needed, you don't have to worry.
Collect Flirt Items
These can be collected from the daily quest rewards, buying Crates, exploring secret locations, or doing campaign missions. Campaign mission 4-4 is good for this, and 2-1 can also give flirt items for newer players.
Trust Missions
Sometimes you may need to run more Trust missions than the five you get each day. Fortunately, you can get around this with nanites. When you start a Trust mission while out of tokens, you will get an offer to refill your Trust Tokens for 50 nanites. This is a small price for another 5 Trust Tokens and avoiding getting stuck on an event task.