There are plenty of ways to earn dilithium without a suit. The dilithium mining mission rewards 400 - 1000 dilithium ore depending how high you score in the mini game. Earning 600 - 800 is pretty easy. Earning the full 1000 takes practice.
As mentioned, the easiest way to get dilithium is the "Path of 2409"quiz at the Academy. You can look up the answers online. There's even an Android app for STO which has the answers as well. It's actually in the "recruitment office" (circular building). Easy 480 dilithium and skill points.
As mentioned, the "Explore New Worlds" missions is a good way to earn 1440 dilithium. You also need to select a mission to explore a sector as well. It can be quick or it can be long. You need to scan anomalies in "unknown" sectors. They can lead to combat or simply scanning mission.
Before getting to level 50 I done PvE mission available at Starbase 24 (same sector as Earth). There's "Defend Starbase 24", "Gorn Mine Mission", and one ground mission that I cannot remember the name. It's not "Breaking the Planet" which seems to be a dead mission. I was never able to up that mission since I attempted to queue it up daily since the beginning of February. I simply ignore it now. You get 480 dilithium each time to do the mission.
At level 44 and above (or is it 50?) you get even more missions available that mostly awards dilithium, but some do not.
Doing DOff missions (Duty Officers) is another way to earn dilithium, but it is a low trickle since you basically earn only 5 dilithium on successful mission, very few offer 50. If you get a critical success, then you can get up to 125 dilithium. However, this does not happen very often until you get better quality DOffs. However, this is a totally different topic. Needless to day, DOff missions should only be considered as a secondary source of dilithium.