My choice, the most important three articles:

• Why truth? And... -- it contains motivation for doing what we do, and explains the "Spock Rationality" misunderstanding
• An Intuitive Explanation of Bayes' Theorem -- a biology/medicine example focusing on women, and an interactive math textbook (great to balance the LW bias: male, sci-fi, computers, impractical philosophy, nonstandard science)
• Why Our Kind Can't Cooperate -- a frequent fail mode of unknowingly trying to reverse stupidity in real life, important for those who hope to have a rational community

then these:

• How to Be Happy -- a lot of low-hanging fruit for a new reader, applying science to everyday life; bonus points for being written by someone else
• Something to Protect -- bringing the motivation to the near mode; the moral aspect of becoming rational
• Well-Kept Gardens Die By Pacifism -- a frequent fail mode of online communities; explanation of the LW moderation system

and then these:

• Making Beliefs Pay Rent (in Anticipated Experiences) -- the difference between a useful and useless belief, and how to avoid discussing mere words
• Knowing About Biases Can Hurt People -- warning about a possible fail mode for people who enjoy reading the articles about (other people's) biases
• Guessing the Teacher's Password -- education is an important topic for many people
• How to Beat Procrastination -- an important topic for many people online, and also very popular one (might bring hyperlinks to LW)

Note: I think that each these articles can be read and understood separately, which in my opinion is good for total newbies. People are expecting short inferential distance, and you must first gain their attention before you can lead them further. If they will enjoy the MicroSequences, they will more likely continue with the Sequences. I also think these articles are not controversial or weird, so they will give a good impression to an outsider. The selection includes math, instrumental rationality, social aspects of rationality.

Funny thing, it was rather painful to reduce my suggested list to only 10 articles, but now I feel happy and satisfied with the result. Please make your own list, independently of this one. (Imagine that you have to select 10 or less articles for your friend.)