I have been 11 times to Argentina and absolutely love it. My favorite combo:
Fly from DFW or Houston to Santiago. Arrive in the morning and take bags to Hotel Aubrey. Have a driver take you on tours/lunch for the day. Dinner reservations at Como Agua Para Chocolate. Fly next morning to BA. Stay for a day. Shop on Florida Avenue, eat at a steakhouse that serves grilled cheese and a nice hotel(I like Four Seasons) Travel over to regional airport for trip to Bariloche. Stay at LLao LLao resort. Spend a day or two shopping, hiking, etc.. Then drive north thru the lakes to San Martin. Stay at Tipiliuke ranch for a few days of fly fishing, hiking, red stag and quail hunting, etc.... Airport in San Martin will get you back to BA.
Have also flown into Delores/Merlo from BA and pigeon hunted and winery thing. Chartered a plane from there to Cafayate and spent a week visiting vineyards, desert, acoustical chamber, and stayed at some really nice resorts. Flew back to BA via Salta. Beautiful country.