I'm having real trouble completing bounty missions. I'm looking for pirates/retros in either two or three locations, so I jump to the first, kill all enemies, jump back, kill all enemies, travel to 2nd location, jump, kill, jump back, kill again. Still doesn't complete my mission. What am I doing wrong?

With some missions if you land before completing the mission then it counts as having failed the mission. Not sure if this holds true for bounty missions, but you can check this by double-checking your mission log after landing and seeing if the mission has disappeared (if it has then you failed it). Bounty missions also tend to be for just a single ship, typically better armed than the normal ships you run into, you just have to travel to the different nav points to find the actual ship that you're after.

The bounty missions are persistent, I have landed several times since accepting the mission, and they're still taking up two of my three mission slots. Am I supposed to fly around each entire system to find a lone ship? I've just been going to jump point, jumping, killing everything at that jump point, then returning. I'll try touring around the system too, then?

ughhhhhhhhhhh :-p

It helps to have a fast ship that can outrun most of the normal enemies. That way you can go to one of the nav points, and if your target isn't there just quickly turn around and head to the next nav point on the list rather than having to duke it out with any other enemies that might be around.