Both big and little cottonwood canyon are great roads to drive on, however they take you to the Utah ski resorts and end. Also they are some what out of your way, as you must go south to get to them. There used to be a road that connected big cottonwood with emigration canyon, but it was 4 wheel drive that last time I tried it, also it may still have snow on it. Hard to tell just where you are coming into Salt lake city, but it looks like you are coming out of Heber city. You will meet up with I-80 west bound you can then pick up Emigration Canyon and it is a great twisty road with lots of switch backs to drive. At the bottom it will put you on Wasatch Blvd, so with a good map you be able to get back on your route through Salt Lake City.