Basic Mechanics
Expedition Events are the redheaded stepchild of DB's events. They're infrequent and by far the most frustrating of events because the rules have changed literally almost every time that they've been run. The following guide is based on the most recent run of expeditions... which may not reflect the next run of expeditions, but should generally cover the gist of it.
What a skirmish event is to ship missions, an expedition is to away missions. The difference is that they are ticket based (similar to cadet missions). Originally, these tickets were timed to one hour. In the most recent expeditions, they've been unlimited - meaning that you can keep grinding away at the same ticket for VP until all of your crew is exhausted. More on crew exhaustion below.
To start the event, you will need to cash in an event ticket. You will get three free tickets per day, resetting at daily reset (i.e. when daily missions reset) and you can purchase extra tickets for dilithium, which will also reset each day.
Unlike in skirmish events, you must work through the various difficulty levels to unlock higher ones - much like an away mission in game (i.e., get all stars on Normal before Elite unlocks, etc.). Once the nodes are unlocked, they stay unlocked for the duration of the event. Difficulty will also scale - just like in away missions - with the highest skill checks on the right and lowest on the left.
Therefore, for the first ticket, concentrate on unlocking as many Epic levels as you can. In each subsequent ticket, because of the crew depreciation (more below), work top down, right to left so you are using your event crew on the hardest nodes first and getting as much use out of them as possible.
Scoring Points
There are 4 rare rewards (star nodes) per mission. Each of these nodes represents a quantity of VP corresponding to the difficulty of the mission.
Normal = 250 VP per node
Elite = 500 VP per node
Epic = 1000 VP per node
Event crew provides a multiplier (1.5x or 2x based on bonus or event crew, respectively) on this VP.
All star nodes completed with variations of event characters will earn you 1.5x the specified VP for that node. All star nodes completed with event characters will earn you 2x the specified VP for that node. Most of the VP, therefore, is in hitting rare rewards on the Epic level. You'll also get a smaller amount of VP for completing the missions as you go, so it's possible to grind out Epic with non-event crew for extra VP, if you're so inclined, but the risk/reward here is not stellar.
Crew Selection
Like an away mission, you’ll choose 3 crew to run each mission, but having a deep bench is very important here.
While leveling event characters is sufficient for faction events, expedition events also require a supporting cast. There are many non-star nodes in the missions, and you need to avoid using valuable event characters for these. You'll also generally require at least 2 passes for each mission. It's often helpful to write down what crew you are using on which nodes and then follow those each run through once you have a rhythm. The good news is, you should be able to max out thresholds in a few tickets.
As it’s easiest shown by example, here’s the run of crew I was using on the last expedition event on my VIP0 account:
Event crew are in gold and bonus crew are in green. Sometimes moving out of order (i.e. doing one pass of 3 Epic first and then doubling back) will enable you to win more VP. You must crit the node with the character in order to get the VP points, so figuring out how to best hit those nodes due to depreciation is key.
Crew Depreciation
Each time a crew member is used, both their traits and max/min proficiencies will drop permanently until the end of the ticket. You will see a small red triangle with a number in it. For event crew, each use will drop stats 5% of the original total. All non event-related characters will drop by 10% with each use. When the number in the red triangle reaches 20, the character will be at all zeroes. Failing a node will cause stats to drop by 50% of original and will add a big penalty to the final skill check. All characters’ stats will be restored to full and the skill checks will be reset when your ticket is completed.
Also, remember hidden skills. All characters have skill points that are added through equipping items as you build them, including for skills which are not listed. Something may add medical stats to Detective Data, for example, even though medical is not listed as one of his skills. During the last expedition, a FE Detective Data was able to crit medical nodes on Normal, even while depreciated.
Final Thoughts
Expeditions seem cheap on the surface because the ticket costs are low, but these events are really about a) your bonus crew and b) the depth of your bench. Using event crew will give you the biggest bang on the VP nodes, so grab a pen and pencil, calculator, and remember that you must be able to crit the nodes in order to get the points before your crew is exhausted. Either way, settle in for a grind, and make sure to write down your optimal crew pathing so you can throw the event on autopilot.