Curious about how race impacts someone's life? Spend time with them. It will become very obvious without you having to ask anything and you'll get the benefit of being able to empathize from seeing it first hand.
Or, if you want to really try hard - work against your biological instinct to segment people into buckets based on physical attributes. Get to know who they see themselves as rather than how you see them by finding out what is important to them.
Here is an easy start: Instead of asking "where are you from" a simple and innocuous question which reveals basically nothing about someone - ask "what kind of hobbies are you into?"