Where should I start troubleshooting this heat pump?

More info may be needed to answer your question. You mentioned holding the condenser fan before turning the stat on. Why? Was it not running and you had to give it a spin to get it going? When the fan was running did it sound normal? If you grab the motor shaft can you move it side to side? Moving the motor shaft up and down a little is ok but side to side is not. Not even a little. That is an indication that the motor is about to lock up.

If it was running fine and just quit and then it ran ok again and quit again, if I’ understand you correctly, then that eliminates many possible problems. From the info I have left it leads me to believe that it is either the cond. fan motor going out or a capacitor out. Most units use a dual capacitor for the fan and compressor but the Trane units sometimes have 2 or 3 capacitors wired in parallel. Look at the capacitors to see if they look swollen. You can take one off and set it on a flat surface. If it wobbles it’s swollen. It should set perfectly flat. That would be the cheapest fix.

If the capacitor(s) isn’t swollen that doesn’t mean it’s ok but it won’t cost much if you want to replace it.

So, we can eliminate several problems: low refrigerant, air flow or anything to do with the air handler or low voltage circuit, including the thermostat.

Check for play in the motor bearings by moving the shaft side to side.

Check the capacitor for burnt wires on the terminals or if it looks swollen.

You can email me if you like and we can get into greater detail.

fan on blower will not start unless you turn with your finger then it run fine until time to restart

before you damage anything more just call a HVAC tech to check it properly