First, as alluded to earlier, learn about state machines. That's really what you have here and diagramming it as a state machine will make it simpler to understand and to program.
Next, I disagree with the speed and floating point comment. If you need floating point, use it. Speed is not an issue with this controller. However, it looks like you're applying a simple form of the Steinhart-Hart equation to reduce the thermistor? If so, don't. You're controlling over a fairly small range, you can interpolate between a couple of fixed points on the resistance-temperature curve and be accurate enough.
So decide on the states this controller will be in. For starters I see OFF, PREHEAT, HEAT_STAGE1 (temp1 == 85), HEAT_STAGE2 (temp2 == 50), VEG_OIL, etc...
Decide on the events that will cause a transition from state to state and then you can build the code as a switch() statement, or my personal preference: a data-driven state machine with states and transition events as a table.
Next, you have a couple of tasks that will run in all states once you leave the OFF state, they will also need to be handled. I suggest you start with the temperature control tasks and get them working first.
I would also suggest you ask this on a programming StackExchange site (my get better ideas) since it's really not about electronics per se.
Best Answer
The easy way: power the Arduino with a separate 5V USB wall wart. Then, use an Arduino relay shield for the 24VAC valves. Finally, use the 24VAC only for the valves.
You can also find 24VAC to USB 5V - search for exactly that and lots of choices come up.
The reason I recommend the USB plug approach is that the losses will be less than using the 9V-ish barrel plug. But you can can do that, too, if that's your preference.