Which tense would be correct in this context?
Since I moved to my parents' old house, I have been going to work by bus.
Is this tense correct? I am currently living in that house, so I believe a perfect tense is the right one, and I assume that it must be progressive since this is happening everyday...?
Present perfect progressive tense refers to an action that has occurred in the past and is continuing in the present and/or will continue into the future. Assuming that is what you intend to convey, the sentence is correct as you have written it.