"Farming" dilithium is a topic which I can not fully describe as it is dependent on the players rank and the progress that he made.
However... earning 8.000 dilithium is not that hard.
There are missions which offer high payouts and there are missions which offer medium payouts in a short time.
Methods (incomplete):
1. There is a mission to repair satellites at the Trealus system of the Eta Eridiani sector block.
about 500 dilithium reward (daily) and the mission is pretty simple and fast (5mins)
Got to the sector south of the sirius sector block and head for the Trealus System (use the system list of your star map and double click the system).
Once there you can enter the system and fly to the first satellite.
Scan it... repair multiple times... head to the 2nd sat and do the same... now head to the 3rd sat and scan it... beam over the repair team.
Now you only need to destroy the incoming ships (easy job) and beam your team back on board.
Thats it.
2. South of the Trealus system should be a freighter called Sh'am (or something like that). Fly to it and hail it.
Now you can play a mission to defend this ship (takes only 5 mins)
500 dilithium reward.
3. Asteroid mining.
Next to Deep Space 9 is an asteroid field. Fly there and beam to the ferengi ship.
There you can buy a spacesuit (just take the cheapest offer).
The Ferengi offers to reward you for mining dilithium from a nearby asteroid.
Open your inventory. Double click the new spacesuit and you will notice that in the middle of your screen you got a new icon. Click it and you activate the spacesuit. Now you can leave the airlock.
On the map you see several highlighted zones. Thats where you have to go and start mining. (with the spacesuit you also got a mining laser).
Go to the crystals there and use the option to mine. Now a mini game starts.
I would recomment to use the arrow keys of your keyboard for this game.
Left+Right will rotate the laser... up and down will size the target.
Task is to put your laser right over the marked dots. every few seconds you will have to rotate and scale your laser marker again.
The higher your score is the better will be your reward.
reach more than 500 points and you can keep 100 of the dilithium for yourself (the ferengi gets the rest)
Over 700 and you get 125 dilithium...
You have to do this mining stuff 5 times on that asteroids so you should be able to get 500-600 dilithium.
4. Sector exploration.
This one is worth about 1500 dilithium (all of the so far methods are daily missions).
I personally would combine this mission with other missions.
In particular there is this general "explore something" mission and there are missions like "explore this star cluster".
Well... the combination i would advice as it seems to be most effective is to take the "explore something" mission and the "explore the Delta Volanis sector" and while you do that you can also accept the "Delta Volanis Diplomatic" mission.
Reason for that is pretty simple - you can combine these three missions and they are very close to the sol system (you can travel there using transwarp).
So you accept these 3 missions... transwarp to sol... leave the system again and head to the delta volanis cluster (left of sol).
Now you start the diplomatic mission (which means to fly to the play.. hail it... see that the planet needs a resource... you offer your help.... if you have the resouce you instantly finish the mission... if not you leave the system and also the star cluster... go back the sirius sector... there you go to the next transport ship (mostly the azura... there you buy 10 pieces of that resource ... go to delta volanis again... click on continue the diplomatic mission and accept to deliver the requested resources)
Now that you have done the diplomatic mission (takes 3 minutes) you will notice that you also finished to explore the first of three star system of the delta volanis cluster.
Now you search for anomalies in that sector and scan them... some of them turn out to be unknown systems... enter them... do the missions.
After doing that 2 times you have explored all 3 unknown systems and finished the "explore the delta volanis cluster"... and with that you also finished the diplomatic mission and oh my god... you also finished the "explore something mission".
Now you can get 3 times a reward. Done.
I assume we have 3000 dilithium now
You can get more dilithium by joining area fights (you have to do 3 fights), you can get dilithium by defending sectors, you can get dilithium by doing story missions, you can get get dilithium by doing pve fleet missions -> "stop the gorn invasion", "find the klingon scout", "attack the romulan temple",....
they should be worth between 500 and 1500 dilithium.
Another important method to get dilithium is to let your crew work for you.
Yep... you are the captain and you have to keep your crew busy.
There are 3 types of crew members.
The first one is the ordinary "red shirt"... every ship has a specific amount of crew members. They keep the ship running and repair in in combat.
The larger you ship is the more crew members you have... in combat they repair damages.
Thats it... more they dont do.
Then you already should have meet your bridge officers... they assist you in every combat with abilities or you can take them to ground missions. They also can lead special areas of your ship... such as security or medical decks or science.
And then you have a 3rd crew type... the Duty Officers. That are crew members with special abilities... such as being specialist in warp technology or being a trader or being a bartender or whatever. They practically are the crew members which lead small teams on the ship.
The Duty Officers (DO) have these abilities for a good reason. You can sent them on a mission. You can tell them fix the warp core, you can tell them cure a person, you can tell them to go on a diplocatic mission. And you get for every mission a reward.
You can get experience... you get special items, you get new crew members and you also get dilithium (so we are back on the topic).
An ordinary mission (short mission take 30mins... longer missions can take up to several days) usually gives you a reward containing at least 5 dilithium.
That is not much especially you can only do 20 "DO" missions at the same time (you can have up to 100 duty officers but for some missions you need more than just one).
20x5 = 100 = not much
Well... thats only half of the truth (actually not even half of it ^^).
First it needs to be said... nobody holds you back from doing 30minute missions and sending the officers right on the next 30 mins mission.
Still that would not be the best dilithium income.
But there is something that can make missions a bit more special.
Depending on the skills of your crew a mission can turn into a critical success... if so then you get 125 dilithium instead of just 5.
10 of these missions (or more)... well i think you can calculate possible rewards for yourself.
Now how to get such special mission rewards?
Every mission has special requirements which are usually display in green fonts. If your crew member has the same ability then the chance that the mission is a success raises... if your crew member has more abilities which fit then the chances raise even more.
There is a diagram showing the chances of success, critical success or even failure of the mission based on the assigned crew members.
It depends on you which members you send to the mission and how the results will be. If you send the wrong guy the diplomatic mission turns into gunfight and the mission fails... in worst case your crew members get wounded or even die.
Important for the missions is also the quality of the duty officers. Just like the bridge officers there are experienced (rare) officers and total noobs.
You start with the noobs... you get your first duty officers at the star fleet academy at the sol system.
The good thing is... you can get better officers by doing missions, by raising ranks and by trading.
You also have the chance to go to the starfleet academy and talk to the 2 officery in the round building. (i advice you to go there regulary as you can get new officers every few days there at the round building and from 2 npcs behind the building + from the vulcan guy at the other side of the area)
There you can exchange 5 normal duty officer (you loose them if you do that) and in return you get 1 uncommon duty officer (which has better skills). You can also turn a few uncommon officers into one officer of the next higher level and so on.
Practically you give a few weak ones to get a strong one.
Task would be to have only rare duty officers at the end (but that takes a long time).
These rare duty officers have better skills and do missions in a much more successful way... which in the end raises your mission reward.
I personally keep 50 duty officers on my ship (enough to do all the misisons)... if I have more I go to the academy and start to change the weaker ones with a few stronger versions.
Hint: There are also holographic duty officers that you can buy for energy credits (check your replicator).
Well... I hope it was not too confusing... I prefered to describe all that stuff because I have some translation problems... I am playing the german version of the game and the translation from the english original to german is already crap (in many areas wrong and often even missing). Translating a wrong translation back to english would cause even more misunderstandings.
If you have question... ask.
If you know more ways to earn dilithium then please share it.
I for sure did more missions which had good rewards but do not remember all of them and currently I am not able to play them anymore since my rank is too high for them.
I also do not think that I have seen much of the game universe yet... it is so fucking huge that it would takes ALOT of time to explore everything.
So there might be better ways to earn dilithium.