T-stat just replaced, I can hear the t-stat click on but heater or ac unit does not come on. What next?

Check for power to furnace (breakers, fuses). Also check for a small fuse in the low voltage wiring in the furnace or on the conrol board. If stat was changed without power to furnace being shut off, you may have shorted some wires during change out and blown the low voltage fuse in the furnace. If your thermostat has batteries, it will click even if furnace has no power. However, if the stat has no batteries, and is clicking, then that suggest you have power to furnace and that all fuses are good, in which case we would have to dig a little deeper to find the problem. Make and model of furnace would be helpful if it comes to that.