Which parts of fresh Fenugreek am I supposed to throw off before attempting to dry them out completely?
I would just pull off all the little stems with leaves on them off the big main stem and leave it at that.

You can remove the yellow leaves if you wish, but there's no harm in leaving them on. I presume once dry you will crush all the leaves and small stems for storage, so leaving the yellow leaves on will not make any difference.

Removing the big main stem will simply help speed up the drying time.

For convenience of moving and turning you can leave them complete, but you should remove the smaller stems with leaves once they're dry, the large stem doesn't generally get used.