Learn Russian: Огромная лодка плывёт сквозь туман.

плывёт shows that the boat is making a single, focused trip in one direction right now. If the boat were sailing around aimlessly in the fog, or if it made this trip regularly as a habit, we would use the multidirectional verb плавает.

I thought плыть meant 'to swim'. Why is it used for a boat?

In Russian, the verbs плыть (unidirectional) and плавать (multidirectional) are used for any movement on or through the water. They cover swimming like a person, sailing like a boat or ship, and even just floating.

What case does the preposition сквозь take, and why doesn't the word туман change its ending?

The preposition сквозь (through) always takes the Accusative case. The word туман (fog) is in the Accusative case here, but because it is a masculine inanimate noun, its Accusative form looks exactly the same as its basic Nominative dictionary form.

What is the difference between сквозь and через if they both translate to 'through'?

While you could say через туман, the word сквозь specifically emphasizes moving through a dense, thick medium or obstacle like fog, smoke, or a thick forest. через is more general and often means simply crossing from one side of something to the other.