If you're very new to TTRPG, I'd agree with starting with the starter. Plus you get both sets of regular and stress dice. If you bought those off Free League's shop here those two dice sets would total about $34 US, so for those and the starter kit, it's just about $10 more. Now the dice aren't essential, you could play with regular 6-sided dice (though ideally two separate pools of two separate colors to differentiate regular and stress dice); but they're cool to have especially if you're a dice fiend. Also cool to have is the poster sized map of the known galaxy (it's regular page sized in the core rulebook, so the poster size is easier to read and just more impressive looking).
Downside as mentioned is that the set is structured to introduce the game to the players and Gamemother through a cinematic scenario with a pre-generated cast. No options for building your own characters. On the other hand, the characters are built with cool tools ("agenda cards") that help/coach the player to role-play the character in a way that players used to playing more "heroic" games may not be accustomed. Also, you don't need the core manual to play the sequel adventure Destroyer of Worlds, nor the forthcoming (think it's already in PDF, not sure why I haven't got my copy yet) Heart of Darkness further sequel.
Hadley's Last Hope in the Core Rulebook"storyline" - wise isn't directly connected to the cinematic story arc of Chariot-Destroyer-Heart (and even those aren't really a connected campaign in the traditional sense, as the players play difference characters, though there are connections in "the big picture"). It's also playtime wise maybe a third to a quarter length of Chariot of the Gods (starter adventure).
I bought the Core Rulebook first. I really enjoyed reading it but had a sort of "this ... is a lot" unease when I thought about presenting the system to my main gaming group (most of whom aren't as experienced gamers as I am); but I wound up able to get the Starter Set at a price that basically meant I was paying for what the dice would be worth alone so went for it, and aside from being stunned by the map, I was very happy to see the set was a well designed introduction to the game system and can attest to its utility as a tutorial before your players dive into making their own characters etc. I'd say it even does a better job that the Core Book scenario, which is a bit more minimal in story.