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To chat is casual. You would chat to a friend when you are having coffee together. You might chat to your mum on the phone about her birthday. You wouldn't "chat" to a police officer :P
To talk is more general. Every time you have a conversation, you're talking. If you say "I had a talk with..." that implies a more serious conversation, however. You would talk to your doctor about a worrying ailment, or have a talk with your child about bad behaviour. But generally, talk only has these connotations in specific contexts.
Finally, to speak is, more often that not, referring to the physical act of speaking. You might say "I spoke to..." but that's quite formal. You might say "I speak with difficulty", "he speaks with an accent"... so you'd commonly use "speak" when talking about speaking itself.
