There are a couple of guides out there that I'm sure someone will pop in and give you a link to but mostly you just want to run a bunch of non-combat Exploration missions. To do this go to any Exploration Cluster (Delta Volanis, Arucanis Arm, etc.) and enter Unknown Systems. If your Tactical Officer pops up to tell you what the mission is about, it's most likely a combat mission (except in the case of Aid the Planet missions, which do award DXP). Warp out and try again. If your Science Officer pops up to tell you the deal, check it out.
You get DXP for any non-combat mission except the Scan for Anomalies ones. Aid the Planet missions reward DXP. Scan the Carnivorous Plants missions reward DXP. Investigate the Murders missions reward DXP. You'll catch on after a while. Good luck and have fun.