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Caveat: I know nothing about space exploration except what I've picked up while playing with Kerbal Space Program, so this could be a terribly stupid or offtopic question.

I'm wondering just how many burns did the Apollo 11 mission require.

Did they first have to reach a circular orbit around Kerbi…–Earth by a single burn at Apogee or did they target the Moon immediately?

Then, when they've reached the Moon did they have to retroburn at Perilune to circularize, or was the trajectory calculated so that it happened automatically?

On their way back was a single burn enough to leave the Moon or did they have to execute some corrections along the way?

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According to the plan, an Apollo mission to the moon uses eleven-ish burns. Here are the expected maneuvers:

Apollo mission profile

  1. Launch from earth. This you might count as three burns, since it requires all three stages of the Saturn V rocket. They're not only launching from earth at this point, but they're also burning to get into a circular orbit around the earth.
  2. Leave earth orbit for the moon (translunar injection). Here they relight the third stage of the Saturn V, using up the rest of its fuel. At this point, they should be on a free-return trajectory, meaning that if they do nothing else, they'll slingshot around the moon and come back to the vicinity of the earth.
  3. Make a midcourse correction. These corrections are made using the engine built onto the Service Module (SM).
  4. Make another midcourse correction, as needed.
  5. Get into an elongated orbit around the moon. This uses the SM engine.
  6. Circularize the orbit around the moon. This also uses the SM engine.
  7. Get into descent orbit. This uses the descent engine on the Lunar Module (LM).
  8. Descend to the moon. This also uses the descent engine on the LM.
  9. Ascend from the moon. This uses the ascent engine on the LM, leaving the descent portion behind.
  10. Leave lunar orbit for the earth (transearth injection). This uses the SM engine.
  11. Make a midcourse correction. This also uses the SM engine.
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In practice, many of the midcourse corrections were either not needed at all or small enough that they were performed with the RCS rather than the SPS. There were also a few separate burns during the lunar ascent phase - one big one to get into lunar orbit, then a couple of smaller ones to set up the rendezvous with the CSM. –  pericynthion Jun 13 '14 at 18:06
Stage 1 and stage 2 could count as burns, as could the initial burn of stage 3. I seem to recall Apollo making at least a couple of orbits around Earth, before the TLI burn -- did they do a circularization burn at apogee, or did they just accept a very elliptical orbit for the short time they'd be parked? –  Phil Perry Jun 13 '14 at 22:22
The S-IVB injected directly into a parking orbit that was close to circular (albeit very low altitude); no need for a circularization burn. –  pericynthion Jun 13 '14 at 23:49

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