First, you are going at the wrong time. It will be busy (and probably warm--if not hot) during July. It would be better if you could go either just before Memorial Day or in September (after Labor Day).
Second, you can't really see much of Yellowstone in a day trip from Big Sky, unless you concentrate on just one area. From Big Sky, your best bets are either Lamar Valley (for wildlife viewing), or the Old Faithful area. You can easily spend a full day walking around the Old Faithful area, and do a side hike to Morning Glory (it is a paved hike, so it isn't difficult, but I bet that most visitors just see Old Faithful go off, walk around to a few adjoining geysers, and then drive off). If your time is limited, avoid the Lake and West Thumb and all of that, and don't head south from the park into the Grand Tetons. You can hit Lamar Valley and/or Old Faithful either coming through the North Entrance or the West Entrance (but from Big Sky, unless you backtrack to Bozeman, the West Entrance would probably be easiest). If you go through the North Entrance, you could hit Mammoth Hot Springs as well.
Expect each stop of the car to get out to see a side attraction will take you at least 30 minutes. Also expect to have difficulty finding parking spaces. The speed limit in the park is 45 (sometimes lower). If you don't stop and see stuff, you don't really get to see Yellowstone.