I replaced the front tires last week - I could not afford to do all four. I have an appointment to replace the back ones on Thursday. This morning my ABS, traction control, and stability lights came on plus my AWD light is flashing. Now I'm worried it's not going to make it to Thursday.
Thursday has passed already, but I'm going to go ahead and try to answer this one anyways.
The reason why your warning lights came on is most likely because the size of your replacement front tires is different from the size of the old ones. More specifically, if the sidewall profile has changed (either from a significantly different tire width, or different profile altogether), then the diameter and circumference of the tire will be different.
This will cause your front wheels to rotate at a different rate than your rear wheels. Now, most AWD cars are designed to allow for this to some extent (ie. during turns), but they are not meant to drive with this condition at all times. Depending on what kind of AWD system your car has, this could have a variety of undesirable consequences - I'm not familiar with the specifics, but it could cause center differential wear, tire wear, or no wear at all depending on what system you have.
Additionally, your ABS, TCS, and ESC lights came on because all of these systems rely on the wheel speed sensors, which actually just measure the rate of rotation for all four wheels. If the speed from your wheels are always different, then I would imagine this would cause problems for these systems.
Long story short, I would recommend that you drive your car minimally during this time, until you can get your rear tires replaced to the same size as your new front tires. You can verify your tire sizes by checking the numbers on the side of the tires. How To Read Tire Sizes | Just Tires