How do you level up a kerbal?

You likely got the star for "Orbit around Kerbin". You can check the experience history of your Kerbals in the astronaut complex by hovering over them with the mouse.

Kerbals gain experience points for visiting different celestial bodies (moons, planets and the sun itself). Each body in the solar system has a different score. The harder to reach, the higher the score.

The score of the celestial body is multiplied with how close the Kerbal got to that body:

• performing fly-by (enter sphere of influence)
• getting into orbit
• "flying" at the planet (low on a sub-orbital trajectory)
• landing
• planting a flag

You get no points for landing and flag-planting on Kerbin (would be too easy, wouldn't it?)

Only the best achievement per body counts for the experience total, and it only counts once. So when a Kerbal was on several missions where he got into Mun orbit, landed, and planted a flag, he will only have 5 points for planting a flag on Mun. But when you get him to do something on a different body, the score of that body is added to his total. So when you want to level him further, take him to another body.

His next step on the career ladder would be exploration of Mun and/or Minmus. A landing and flag-plant on either should get him to level 2.

Each Kerbal gains XP, which is used to level up. This chart shows the amount needed to reach each level. A Kerbal earns XP when it returns from a mission where it did something new such as had it's first flight or planted a flag. A chart showing all the amounts for each activity on each Celestial body is available here. A kerbal can only earn one lot of XP per Celestial body so planting a flag on the Mun will earn you 5 XP (even though you will have orbited the Mun and landed on it which also gives XP. You only get the highest thing that you do).