To start, I want it to have a thermostat on the touchscreen
I don't see that this is possible. If you put a thermostat on the touch screen, it will interfere with seeing and using the touch screen.
A user interface that allows setting high and low temperatures is a different story.
Clear requirements lead to clear code. Fuzzy requirements do not.
How would I program the arduino to do two things at once(cycling the machine and controlling temperature)?
You can't. You can program it to do two different things very close together, but there will be a distinct interval. Whether that interval matters is unknown, without knowing how you are "controlling temperature" and what "cycling the machine" means.
From the tutorial link:
Uses digital pins 5-13 and analog 0-3. That means you can use digital pins 2, 3 and analog 4 and 5. Pin 12 is available if not using the microSD
That's not many pins for the stuff you want to do.
I would have suggested a Mega, but:
The shield is fully assembled, tested and ready to go. No wiring, no soldering! Simply plug it in and load up our library - you'll have it running in under 10 minutes! Works best with any classic Arduino (UNO/Duemilanove/Diecimila). This shield does work with the Mega Arduinos but its going to be half the speed of the Uno-type boards because of the way the Mega rearranges all the pins (there is no way to get around this!) This shield is not Leonardo-compatible
Whether "half the speed" matters, or not, is up to you to decide, based on no data.