I am trying to understand what is the best way to find correlation between the contracts composing the forward curve. Is it just lining up the front month, front +1 month contracts and so on, calculating their price returns and finding the correlation of returns or is there more to it?

That sounds like a reasonable starting point. Beyond that it depends a bit on which market you are looking at. It may make sense to do a principal component analysis of the correlation (or covariance) matrix and look at the factor structure. If it is a storable commodity there will presumably be a  dominant factor driven by the spot price and the remaining patterns in correlation arising from moves in interest rates, storage cost, and market technicals. If its something essentially non-storable like electricity, interest rates or VIX, then the story gets more complicated.