What does it mean if you start rolling down a hill while in drive?

Okay… there’s a problem with my car. I think. And yes, I’m taking it to a mechanic, but I’m curious in the meantime, plus I like to be informed.

Today when I started driving my transmission, which is an automatic, seemed to be acting funny. When I put it in drive, and then pressed down on the gas, it took 2–3 seconds where it did nothing (as if I was in neutral) before shooting forward with a little lurch. This happened about 3 more times, while at lights. The light would turn green, I would step on the gas and… nothing for a few seconds, then a lurch and off we go. Once, while at a stoplight on a hill, I actually started rolling backwards while I was pressing on the gas.
The only thing that seemed to help was pressing down on the gas very slowly and gently as I started. And I already checked the transmission fluid, so I know it’s not that.

Any ideas or suggestions? Thanks.