What I'm trying to do is find a way to cut out the snake in some places so that the sword can show up and it can appear as if the snake is coiling around the sword. I've tried numerous techniques but nothing seems to be working..
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Whenever possible I prefer to work in a non-destructive manner to make future editing much easier. To this end I, personally, would not use Pathfinder here. I'd use a Clipping Mask on the blade.
Object > Clipping Mask > Makefrom the menu
You could alternatively mask the snake, but the blade is a simpler shape. So, if edits to the blade are made later, repeating this process is easy. If edits or more detail is added to the snake, and it's masked, repeating these steps - while still relatively straight-forward - could be a bit more cumbersome.
This is a non-destructive set up, allowing you to retain the shapes in tact and merely change what is seen. Pathfinder would be destructive, making future edits much more difficult. You can always "bake in" the clipping mask later if needed by flattening the artwork or expanding and then using pathfinder to merge objects.
If you do not wish to have a clipping mask in the file. I would still set things up this way. Then at the end, one final step -- select all and Click the
Merge button on the Pathfinder Panel to flatten the objects.