I'll try diplomacy, but if that starts to become a grind, I'll get bored soon.
The best thing you can do is combine it with your Dailies, or Accolade hunt. Since the "Rescue Deferi Captives" grants you 10 DXP pr day, thats a good start.
Most missions grant you another 10 DXP (Scan anomalies (< be aware that the space scan, where you hunt Reshearch Anomalies does not count toward this), Aid the Planet, Discover ships fate, and also the "Get station operational"/"Find out what happend on station") all grant 10, and once you've collected 720 DXP you get access to First Contact missions, where you have to go meet another race, collect info about them, and once you're done, you have to be able to respond "correctly" to the leader. If you pass this puzzle, you get 20 DXP... If you fail, you get 10.
Anyways: Hunting for theese missions specifically will give you your emblems / accolades, and dosen't make it feel (as much of a) grind.