2006 Audi A3 drivetrain & axles problems

Was a normal day of driving, car was warming up. I live in florida so it warms up pretty quickly. I was pulling out onto 4th street when I gave the car a lot of gas, but the transmission did nothing. It just revved up and then after about 1 or 2 seconds, the transmission engaged and the car began to slowly accelerate. After this 1-3 second delay, the car was driving like normal. But the initial hesitation almost caused me to be rear ended because I was pulling into traffic.

From a stopped position on a side street in a left hand turn lane I stepped on the gas pedal and the car started to move then immediately had a loss of power in making a left hand turn onto a major street. The car lurched forward then hesitated. It is really dangerous because it can surprise people and cause cars to crash into me. It has done this a few other times. I feel given the right circumstances this could cause an accident.