When my AC turns on, the compressor does not start immediately. Is my thermostat configured incorrectly?

Your new Trane furnace is a very reliable and efficient furnace, but all it does it move the air for the cooling function. Your Goodman outdoor unit is a 3 ton standard efficiency condensor, most likely made for use as an R-22 system. The indoor evaporator coil is mounted at the outlet of the furnace so that air from the indoor blower will pass across that indoor coil. R-22 systems have been the backbone of residential comfort cooling for many years, but it has been banned and is no longer manufactured. It is still available for purchase until existing stockpiles run out or 5 years goes by. R-410a has been developed as the replacement for R-22, but they are not compatible refrigerants. They can not be mixed. The electrical motor of the compressor has insulation in it to prevent short circuits. The refrigerants perform the same jobs, but they perform at different pressures. The "old" R-22 systems are not designed to operate at the higher pressures "new" R-410a will develop. A coil designed to operate with R-410a will be capable of operating with R-22 in it. A separate issue comes to mind though. Both refrigerants have to be pumped by the compressor. The compressor is a mechanical device that requires lubrication to reduce friction as it operates. R-22 systems use a mineral oil which mixes well with the R-22 refrigerant. The insulation on the windings of the compressor is compatible with the R-22 and the mineral oil. All is good. Change the R-22 to R-410a and the refrigerant and the oil will not mix. The refrigerant will work with the oil to attack the insulation on the windings, in effect burning out the compressor. Any coil that has had R-22 circulated through it will have to be thoroughly flushed clean before it can be used with R-410a in it. If the indoor coil is designed for R-410a, you should use it with R-410a. Just because you replaced the leaky evaporator coil, doesn't mean all the leaks were found and fixed or that new ones haven't appeared elsewhere. You might want to have the charge checked to make sure it's not leaking again/still.

As for the fan starting and time going by before the compressor starts, no, that is not normal. The outdoor unit has a single contactor that allows power to the compressor and the condensor fan at the same time. They should start at the same time.Some systems come with built in time delays that will prevent the outdoor unit from being started until a certain length of time has passed, typically 3 to 5 minutes ( heat pumps always have a builtin delay on the defrost control board), but none that I know of allow the fan to start and delay the compressor. There are some commercial type systems that delay the compressor startup, but they use different types of controlling systems than residential comfort cooling and it is doubtful you have that.

Your best bet may be to go ahead and replace the ourdoor unit and get a new R-410a system in place. Typical lifespans are reaching 10 to 12 years now for new units. They perform very well and with excellent efficiencies for long periods of time.

Food for thought. If you are planning on remaining in this home for a long time, you might want to consider having a geothermal heat pump installed. Costly up front, but great systems, and they pay for themselves rather quickly.

there is a time delay built into the software on the defrost board. This is a safety measure. Rest assured this is fine.