How do I use "speak to you and speak with you?"

Answer 1

You can think of those phrases in terms of one-way and two-way communication. "Speak to you" implies a one-way communication, i.e. "His father spoke to David about his poor grades." (David does not necessarily speak.) "Speak with you" implies two-way communication, i.e., "I would like to speak with you about your opinions on global warming." (I would like to hear your opinions.)

Answer 2

When thinking of the differences in these very similar phrases, you want to consider the simple translation and go from there...essentially they mean the same thing as they do in English. Hablo a ti vs Hablo contigo mean the same message but offer two different grammatically accurate ways of communicating that thought. Though both are correct, contigo is more commonly used in my experience when speaking with someone verbally.