How do I ask a question politely?

When I was growing up, if I ever said something similar to "Can I go to the store with Joe?", my mom would correct me with "May I go to the store with Joe?".

Is "May I?" the typical way to ask a question politely or is this just specific to where I grew up or what my mom learned?

As I learned it, "may" is about permission while "can" is about ability.

• "May we borrow your car?"
• "Can you say 'Irish wristwatch'?"

So your mother was correcting "Do I have the ability to go to the store with Joe" to "Do I have permission to go to the store..."

I agree these are largely interchangeable.

You can take it one step further and employ the subjunctive mood to make questions more polite. These work well if you want something of someone, rather than mere information.

• Might I join you?
• Could you please say this?

In the last example, though, note that could can't really be replaced with might, let alone may. Would is also appropriate.

I don't think it matters if you use "may" or "can" - both can be polite or impolite. This is because politeness has as much to do with intonation and context as with the actual words used to build the question. The relationship between the interactants is also key.