Learn Farsi: خانه‌ی ما سمت راست است.

The preposition be ('to' or 'towards') is used when there is movement, like walking towards a place. Because this sentence describes a fixed location ('is on the right'), we drop the be.

Do I need a Farsi word for 'on' to say 'on the right side'?

No, Farsi usually skips positional prepositions when talking about right and left. samt-e rāst literally means 'side of right', which naturally acts as 'on the right side' all by itself.

Why is there an extra letter ye (ی) at the end of khāne (خانه)?

This is the ezāfe linking 'house' to 'our'. Because khāne ends in a silent 'h' (which makes an 'e' sound), you must add a 'y' sound to smoothly connect it to the next word. In writing, this is shown by adding the letter ye (ی), making it khāne-ye mā.

Could I use taraf-e instead of samt-e here?

Yes, absolutely! samt and taraf both mean 'side' or 'direction' and are very often interchangeable. You could say khāne-ye mā taraf-e rāst ast and it would mean exactly the same thing.