I was stopped at a red light in the outermost lane with two other cars in front of me, and when we started moving a car from the middle lane made a lane change and then had to brake to avoid the car that was originally in front of me. He didn't turn on this signal until halfway into the lane change. I started braking as soon as I saw him moving towards my lane but there wasn't enough time or distance to stop my car.

Am I at fault because it was a rear-ending accident, or does it make a difference that he made an unsafe lane change? I had no witnesses and he had another friend in the car.
What will happen to my insurance? How will they deal with this?

Did you call your respective insurance companies? An accident investigator will need to look at both cars as they are trained to be able to determine fault based on the damages. A rear-end accident is not always 100% of the time the fault of the 2nd driver. This sounds like you may not be at fault but an accident investigator can say for sure. Several factors could cause the situation to change (ie. Angle of impact).

As for the impact on your insurance, that depends on what state you're in and the insurance laws there. Your prior driving history will be considered when it comes to affecting the premium.