Learn Russian: Вы хотите идти тайгой или болотом? - Do you want to go through the taiga or through the swamp?

This is the Bare Instrumental of Route. It is used to describe moving directly through or along a continuous space, turning the environment itself into your path. It translates roughly to 'by way of' or 'through'.

How does using the instrumental here feel different from saying через тайгу or по тайге?

Через тайгу implies crossing from one boundary to the other to get past it, while по тайге suggests moving around within it. The bare instrumental (тайгой) focuses on the terrain acting as the route itself. This structure is very common when talking about natural landscapes like forests, fields, or seas.

Why use the unidirectional verb идти here instead of the multidirectional ходить?

The speaker is asking about a specific, single trip from a starting point to a destination that is about to happen. Because it is a planned, one-way journey, the unidirectional идти is the correct choice.

Болото is a neuter noun, so why does it take the ending -ом in болотом?

In the instrumental case, neuter nouns share the exact same endings as masculine nouns (-ом for hard stems, -ем for soft stems). So the neuter noun болото simply becomes болотом, just like a hard-stem masculine noun would.