"Speaking with" someone means having a conversation.
You can also use the preposition "to" after "speak". This is more appropriate when you're talking to an audience that's not speaking back to you very much. For example, you use "speak to" when someone talks to a large group:
My kid's teacher asked me to speak to the class about my experiences as an emergency room nurse.
Use the preposition "about" to introduce the topic of the conversation:
I spoke with her about her tardiness. She assured me that it would not happen again.
"Speak" and "talk" mean the same thing. "Speak" sounds just a little more formal, though.