Thanks for your replies and my apologies for not researching enough on my own. I did search and read 4-5 posts about this before posting but was somewhat confused.
I have been doing some more reading. The fan being a 4-pin simplified things a bit.
So if I understand this correctly, this is all I need/want:
Arduino Uno (ordered)
Breadboard (ordered)
Button (ordered)
Wires (ordered)
LCD + 220ohm resistor
Tempsensor
12V, 1A power source (already have one)
I will need to set the PWM frequency to 25kHz
Lowering the voltage going into the Arduino may reduce some heat. Maybe put some diodes in serial there.
Unless I have overlooked something?