What came first: the Sun shining or the existence of helium? This seems to be a chicken and egg problem, the sun begins shining due to hydrogen becoming helium, but it's odd that there was no helium initially without the stars.

Is my logic flawed?

(Note: First year astronomy student)
The Big Bang theory predicts that (depending what assumptions you choose) the initial elements were formed from 10 seconds to 20 minutes after the Big Bang.

The initial elements were 


mostly hydrogen
some deuterium 
some helium-3 and helium-4
a little lithium-7
a couple of unstable isotopes that decayed to lithium-7 or helium-3.


As the linked Wikipedia article says


  Essentially all of the elements that are heavier than lithium and beryllium were created much later, by stellar nucleosynthesis in evolving and exploding stars.


So the answer to your question is: the Helium came first.