How does size affect temperature of Sun vs. Earth?

The Sun only has one way of emitting heat, through nuclear fusion. The size of the Sun has nothing to do with how it can achieve higher temperatures, it's all about the mass of the Sun.

The more massive a star is, the more heat and higher temperatures it produces. The Pistol Star is more massive than VY Canis Majoris, but smaller in size, yet emits more heat than it does.

As for planets, even if the Earth were bigger, we would see no noticeable heat change. We might see a drop of heat actually, since our core might cool down faster.

The sun does not have "few ways to generate heat". At its core is a natural fusion reactor producing huge amounts of energy. It's like billions of hydrogen bombs exploding every second. That's where all the sunlight you see comes from. SIze does relate to temperature, however. If you squeeze something into a smaller volume, such as a gas, it heats up. Then it cools down again if you expand it. That's why refrigerators have a compressor.

Planets do not generate their own luminosity and stars like our sun heat up 10% every 1b years

http://science.howstuffworks.com/sun2.htm