AC units and thermostats have built-in delay features when they're shut down and then repowered. The delay can be as long as 10 minutes. And, if you've subscribed to an energy-saving device from your local power utility, the unit can take even longer to reset. If you've installed the parts shown and reinstalled the disconnect block, repowered the circuit breaker, turned on the switch at the furnace, moved the thermostat to AC mode and lowered the temperature below the indoor temperature, and the unit doesn't fire up after 30 minutes, it's time to call a pro.
The AC contactor (relay) and start/run capacitor(s) (see illustration below) fail most often and are inexpensive. So it's a safe bet to buy and install those parts right away, especially if your air conditioning service unit is older than five years. The condenser fan motor can also fail, but it runs about $150 — hold off buying that unless you're sure that's the culprit.
"Customer service was great, and the price was in-line with other places, and fast service. We had a whole replacement of the outside and inside. I would have liked to see a better install for the furnace and new thermostat - the new thermostat should have gone where the old one was, and even tho the new furnace was a tight fit, they didn't offer to come back and repair the case molding, or give me a discount on the damage done, and I believe they should have put a disconnect box on the outside of the home, to make it easier for any future repairs needed. Other than that, am very happy with the cooling and heating!"