I recently installed a new honeywell thermostat because the old one wasn't measuring the temp correctly or consistently. The new thermostat is doing a wonderful job of maintaining a consistent indoor temp but whenever the cooling cycle begins the ac unit starts up then there is a few second delay followed by the fan kicking in. About 20-30 seconds later the fan shuts off for about 3 seconds then starts again and works fine throughout the rest of the cooling cycle.

Any ideas why this is happening or how to fix it?

The new thermostat should have nothing to do with the blower. Power is supplied to the blower circuit the same time as to the A/C compressor relay. However the blower has additional limit switches or timers controlling the relay for the fan. The problem could be in some of the other switches or the fan relay itself. It may be designed to delay start up or shut off the blower. Finding a schematic of the controls would make it much easier to figure out.