A couple disclaimers before I start this. For one, I'm assuming you've watched Stay Night, or Unlimited Blade Works already. If not, watch one of those first. I'd recommend Stay Night, but at least one of them is essential before you watch this. Also I'm reviewing both parts of Fate Zero, cause they're the same storyline and I don't wanna type the same thing twice.
So, you've seen Shirou's storyline, run into the plot twist, and now you're ready to see the grand story of his father. Which is unfortunate, cause of the only major failings of Zero is that it doesn't really touch on Papa Emiya very much. His story is still told, and there's some really strong story beats towards the end of Zero, but for the most part Emiya is on the side, probably brooding and being edgy.
Don't take that to mean Zero is bad though. The only reason Emiya gets sidelined is cause there's so many other plotlines and character arcs to get through. From characters we've already met, like Saber and the Tosakas, to characters we'll really get to know later, like Waver, to characters just featured here, like Zero's Lancer, there is tons to see in Zero. And because all these plotlines are interwoven around each other, no one part feels limited.
Which is what makes Fate as a series stand out so strongly. Getting to see the magician houses throughout their history, seeing the lifelong growth of characters, and watching Saber rock a black suit all make Zero feel grand, and truly like we're getting to see the backstory to an even grander tale. Which might feel weird coming from Stay Night, but trust me, if you're heading to UBW after this you've got so much good ahead of you.
Tie it all up nicely with some incredible music and animation that makes me jaw drop even six years after release, Fate Zero is a powerhouse of a series. Watch Stay Night, knowing that Zero's going to make it all worthwhile.