What does the third-order intercept point (IP3) tell me?
Well, IP3 is a figure of Merit for comparing Amplifiers. Higher IP3 values indicate that the amplifier is more linear, and will generate fewer output tones, otherwise known as intermodulation products. Most radio physical level protocols have IP3 requirements as part of the receiver specifications. For example, GSM requires that the receiver sensitivity degrade only 3 dB with 2 interfering signals at +800 kHz and +1.6 MHz from the desired signal, and at a power level of -49 dBm. These two tones will create an intermodulation product on channel, interfering with the signal you're trying to receive. The higher the IP3 performance of the receiver, the lower the interfering signal. 

More details about your application would help with more detailed answers. 

At the 1 dB compression point, the gain has dropped by 1 dB. Putting more power into the input at higher levels than P1dB cause even more gain drops, and you'll run into output power limitations of the amplifiers. Spectrally, Spurious and intermodulation products increase at P1dB, and you'll see IP5/IP7 tones start to show up.