The merchants that require badges on Qo'nos required you to do Empire Defense missions, or at least did in the Open Beta. There should be several Empire Defense missions of staggering difficulty (meaning I would not go there until you get a mission to do so, as they are probably scaled to certain levels). The grade of the badge you get is equal to your rank. The badges you get for PVP are turned in at Galdena (?!) a planet just inside the Klingons portion of the Neutral Zone. Again same thing with the grade of the badges. The rank for badge grade works the same for the Fed. If they are adding more PVE content later, then I'm sure you can the Exploration badges from those missions too.
You get the first order exploration badges from doing the quest that requires you to complete the Khaless Expanse (kill 10 patrols) three times when you are level 6-10. When you are level 11-20 you will receive the second order exploration badges for doing the same quest. You get 75 each time you complete it. Currently we cannot receive T3+ exploration badges because there is no quest tied to the level 21+ pve space system (Azure Nebula).
You can also go to Ganalda in the neutral zone (Eta Eridani sector block) to spend your PvP medals on similar quality gear (greens/blues).
Edit - We do have ground PvE too on Teroka which rewards a nice ground item to people with high scores at the end of it... It takes a while to complete and you don't get many skill points doing it though. You can also PvE/PvP at N'Vak (KvKvBorg) and Ker'rat (KvFvBorg) which drop good items for top scoring people. There is also another space PvE system named Yov'Bot that is supposed to give a nice drop if you finish it. Fastest leveling is PvP though, followed by doing Khaless Expanse mission repeatedly.