Learn Icelandic: Síðan færir þú peðið.

This is the V2 (verb-second) rule in action! Because the sentence starts with the sequence adverb Síðan, the verb (færir) must be the second piece of information. The subject (þú) gets pushed to the third spot.

Could I use svo instead of síðan in this sentence?

Yes, absolutely! Svo and síðan both mean 'then' or 'after that' when listing a sequence of events. You could just as easily say Svo færir þú peðið. Síðan can sometimes sound a tiny bit more deliberate, but they are interchangeable here.

Does the verb færa work for any kind of 'moving'?

Not quite. Færa specifically means to move, shift, or hand an object from one place to another, like a chess piece on a board. If you were moving to a new house, you would use flytja, and if you were physically moving your body, you would use hreyfa sig.

How is the word peðið formed, and what case is it in?

It is the neuter noun peð (a pawn) with the definite article -ið (the) attached to the end. It is in the accusative case here because it is the direct object receiving the action of being moved. For neuter nouns, the accusative form looks exactly the same as the nominative (dictionary) form.