To slog one's way(or variations thereof) means to make difficult progress forward against significant resistance. To use this as an idiom is simply to apply it to things that are not physical.
Example: "Slowly, painfully and reluctantly, congressional Democrats are slogging their way toward acceptance of President Barack Obama's tax cut compromise, which would let rich and poor Americans keep Bush-era tax cuts that were scheduled to expire this month."
Here, "toward" is used because there is a specific figurative destination ("acceptance") rather than simply a direction, such as "ahead." (Which would then be "slogging their way ahead," but that is not what we see here.)