My bathroom tub gurgles when I have water draining from the bathroom sink.. In addition, I will occasionally get some black specks that will burp out of the sinks overflow drain hole.. Any suggestions on where to start looking to correct these problems?

A gurgle tells me that your lavatory's attemptingto vent through the bathtub trap.
This indicates a blocked vent. The solution would be to snake from the lavatory roof vent.
Pot out enough snake to reach the base and 20 feet more. Good luck, Tom

Agree, most likely cause. What happens when you fill the tub and drain it all at once? Watch the toilet bowl level and the water in the sink trap.