New to Arduino, Have what might be an over ambitious project NEED HELP!

NSullivan:
Thank you for answering the variable voltage question! I'm sure when I get deeper into this, I'll have a bunch more questions on that line.

My thought was to use disposable or rechargeable batteries to power the heating coils, not power them off the Arduino. What I'm not understanding is if there's a way to control the voltage in that system, to maintain 3.7 volts or thereabouts. I know I'll have to wire the grounds together. Would the PWM solution work there? I know people use secondary power for motors and servos and such...

As far as the 20 mA limit, I have literally no idea. I studied electrical systems years ago, but I've lost most of it. This is all the information I have: The battery I'm using to power one heating element now is 3.7v, 1100mAh, and the element has a resistance of 1.8 ohms. This is a preassembled unit, though, and using it for my project would be difficult and possibly dangerous(Lithium Ion and I have a bad history).

I'll assume that the 1.8 Ohms is the cold resistance of the heater. By Ohms law: Current = Volts/Resistance. The current drawn by one heater initially will be 3.7/1.8 =~2Amps. This will change as it heats up. Far too much for an Arduino pin to supply directly, but you could use a transistor.
I'd be looking for a powered solution instead of rechargeable batteries, but that's my preference.
But won't the vapour condense in the tube before it reaches the 'smoker'?