After this mission though, I really need to start getting some dilithium. Not sure yet how that works, so I'll look into that this weekend.
for all the ways, but for a beginner some easy ways are:
-Do the "Explore Strange New Worlds" mission. You probably take the "Explore [whatever] Cluster" mission anyways, since it's available and easy with some decent XP awards, so when you do this you get a bunch of dilithium to boot. And it's daily, too.
-Do the "Investigate Officer Reports (daily)" mission. You have to do three Foundry missions, but some folks have made some short Foundry missions like the ones you would find in the "Explore Cluster" missions, so this can go pretty quick. You can also do the longer missions, some of which are better than any of the official missions.
-Asteroid mining can get you a few hundred more dilithium, risk-free, every day, plus some random particle traces and stuff if you are good.
This will build up your dilithium stocks slowly (you won't get 8000/day this way), but if you do it every day it adds up, and it's easy.
Of course, the easiest way is to sell Zen, but that costs real money, and it looks like the D/Z exchange rate is going down.
At Starfleet Academy, in the northmost building, you can do the mid-term daily. It's answering a question about the game history up to current day. It takes no more than a minute or two, and pays 480 dilithium. And all the answers to the test can be found at http://www.stowiki.org/Lore
There are also three investigate officer reports at the academy. Just search for 'academy quickie' The location of those missions is outside the building where you took the test, by the water.
In less than five minutes you have almost 2000 dilithium.
Do those when you are under a time crunch for your dil. But you really should do a selection of foundry missions, many are excellent.