My carrier HVAC system does the following. When it first comes on it cool just fine, but when it goes through a on/of cycle tripped by the thermostat the blower comes on, the outdoor fan comes on but the compressor does not. If I turn the AC off using the tstat, then turn it back on, after about 15 minutes, everything works fine again. An Ideas where I should start looking to find the problem so I can fix it?

Sounds like the compressor may be short cycling and not restarting, may need a hardstart kit, or to clean your outdoor unit.

I would make sure the condenser coil is clean as joshdata said and install a low voltage time delay at the condenser