The dog will jump in the air to try to catch them and sometimes chases them from room to room. She actually caught one yesterday and was eating it (something that cats tend to do). The cat will chase moths in the garage, but doesn't appear to have much interest in flies (she did when she was younger).
When I encounter flies, I take a paper towel and try to capture them within it. Once I am able to do so, which can take some time, I open the deck door and let them fly away. Sometimes (though not too often), I'll clip a bit of their wing when I envelope them, but that's certainly better than what would happen if they'd wandered into virtually any other house in the neighborhood. My wife will sometimes get frustrated at my strategy saying things like, "Just kill it", but it's quite obvious to me that the fly just wants to be free; if a little extra effort can ensure this, it's a win-win.