Had a similar situation.
A Certain Magical Index came out with a "Side story" series revolving around Mikoto Misaka, which was taking place before the events of Index, it's called A Certain Scientific Railgun. I was panicking about whether I should watch Railgun or Index first, and found out I was freaking out for no reason, because it doesn't alter anything in any way.
I suggest you watch Fate/Stay Night and then watch Fate/Zero as I watched Index and then Railgun, because it feels like you will catch more things inside of the prequels than you would watching the prequels first.