Ironically, Yu-Gi-Oh! is the game with the obscure chain. No other card game has unclear priority and the back ad forth chain that we have. But Yu-Gi- players have goten used to it, and the VS chain seems odd, even hard to grasp at times.
It's actually quite simple. If you have the intiative, then you get to go first at each phase or pass.
Starting a chain (which just about everything in this game does) does not mean your opponent gets to respond. You build your chain first. Once you've decided your done buildig the cain you pass to your opponent.
He now has priority to build onto the chain. And he gets to do the exact same thing. He adds effects and you don't respond until he passes priority back to you.
If two players pass in succession then the chain begins to resolve.
The chain resolves a little differently. Yes, it resolves backward with the last card played, just like in Yu-Gi. But...
Say, I've built a cahin with two effects on it. I pass to you and you put two more effects on the chain. Then you pass back to me. I now have the option adding on or passing. I choose to pass, so the chain begins to resolve.
But when the first effect resolves, a player may add an additional effect onto the existing chain and the whole process of passing back and forth can continue like before.
Basically, the chain can up and down before it ever fully resolves. As efffects resolve off of it, new effecst can beadded onto it, like a stack of blocks.