The IAT

I have found the Implicit Attitude Test (IAT) to be useful in starting a conversation about one's awareness around issues of race. What is the IAT? From their website:

It is well known that people don't always 'speak their minds', and it is suspected that people don't always 'know their minds'. Understanding such divergences is important to scientific psychology.
This web site presents a method that demonstrates the conscious-unconscious divergences much more convincingly than has been possible with previous methods. This new method is called the Implicit Association Test, or IAT for short.
In addition, this site contains various related information. The value of this information may be greatest if you try at least one test first...

Here is the link to the IAT. I'd recommend starting with the Race IAT.