First and foremost, if there are refrigeration service techs on the forum, feel free to jump in. I am not a refrigeration tech.
Looking at page 6-18 the contactor seems to be the only thing that would keep both the fan and compressor from starting on power up. Since you say you hear a click (something other than the switch), the fuse and transformer (both appear to be on the control board) should be OK. Now you need a decent multimeter and the know how to use it safely. I recommend an insulated alligator clip. Some of what follows has to be done with the unit powered. If you are not comfortable with any of it, walk away.
With power applied and the unit turned on, if you read any voltage (probably around 24V, possibly around 12V) between TB30 and TB37, the contactor coil, fuse and transformer are OK. If not, check the fuse. If the fuse is OK and you see voltage between TB32 and TB35, the high pressure cutout is open.
If you read line voltage between TB35 and TB33, the contactor is probably bad.
With power off and the unit unplugged, disconnect wire 42 from TB35 and use the insulated alligator clip to connect the wire to one of your meter leads. Your meter leads will have 120V on them so beextra careful. Plug the unit in and turn it on. If resistance is at or near infinity, the contacts in the contactor are bad. Turn the unit off, unplug it and reconnect the wire.
With the unit still unplugged, disconnect wire 74 from TB30 and wire 73 from TB 37. Check the resistance between wires 73 and 74 and if it's at infinity, the contactor coil is bad. Reconnect the wires.
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I don't want to get anyone hurt and since I never know the reader's capability, I probably go overboard in keeping it simple.