I'm new to 4x4 service, but I'm curious, what is the best way to refill the transfer case? Use of a funnel or is there some other method?

There are many ways to do it, one method that works well for most people, is to connect a piece of 3/8" rubber hose about 4' long to a funnel. Remove the fill plug, stick the hose in the hole, and have an assistant hold the funnel and dump fluid into it while standing next to the vehicle. When fluid runs out of the fill hole, put your thumb over the end of the hose, and put the plug back in.

On that one if a 136-236-246 transfer case.
Be sure to use the GM auto-trac transmission fluid.
There is a tsb someplace on them
If high mileage or acting up some will need a flush.
Drain and fill and drive and then drain and refill again.
MT