Looking for help with some peculiar symptoms Ive observed in my 2013 Silverado 1500

[Mfire123]

I own a 2013 Silverado 1500 regular cab 5.3 V8 4x4 with 56500 miles on it. I had a lot of gremlins pop up in it that went away when I finally replaced the battery. One thing that didnt go away, I start it up and drive it anywhere from 10-50 miles (havent gone longer than that) where it runs great and smooth and then I reach my destination and park it and shut it off. It will sit for 30 minutes, maybe a little more or a little less. Then I get back in the truck and fire it up and drive it back towards home and Ill notice that it runs rough. It feels "bouncy" with a steady vibration and the fuel mileage is still below what it should be getting. It isnt just when braking or when taking corners or anything. It runs smooth on the way there, and runs crappy on the way back. I only drive it once a week and thats usually just to cycle it through the motions. Any idea where I could start diagnosing? 

[swathdiver]

You need a good scan tool Mike.  Essential to work on these things.  If it were mine the first place I'd look is the fuel trims to see if there is a vacuum leak at the intake gaskets.  Would also check out the fuel pump and injectors and coils.