Who archer should invest first?

your Supports and Playstyle

First, all Archers deserved invest since all of them are quite stronk in FGO world.

Orion worked the-second best level against male servants (next to Euryale), meanwhile Orion "have" very tough skill combo and one best NP debuff effect to enemy, excellent self debuff-resistance, when facing to a challenge Boss, Artemis can focus on very durable fight.

Kuro is more focusing on bust her damage in few rounds. She can do NP spam if Critical star is enough and have Mozart/Tamamo as side. Unlike Orion, she can slash out Boss in a short turns, but she worked somehow not decent in durable long-tough fight since no strong-debuff to enemy as Orion's NP, and almost no team-support.

Comparatively, since your Orion is NP1, then I think Orion worked better for stall long-fighting (Art's NP spam for team), and Kuro worked better for short-busting fight in few rounds (Self NP Spam and Brave chain). Off course it is still okay to have Orion as Critical busting damager if you pair them with 4060 dual Waver and have Orion's Quick card more effective; and Kuro can also worked in long-fighting with Tamamo.

Emiya on the other hand is more focusing on Critical damage right now. Even though he has 3 Art cards is good for NP spam, somehow he is not good for NP damage due to AOE and mediocre side-effect (only defense ignore and attack reduction). Then it is a good idea to have him with Waver/Hans/Merlin for providing Critical-up, or Helena/Marie Caster for providing Enough Star to him (and Merlin/Hans worked same). 100% critical up is not kidding.

Chloe's 3rd skill should be shot up to lv10 asap, 50% of a charge every 6 turns along with a ridiculous star gen bonus, then probs Orion's skills as she only really has 1 job, punch men in the junk, to do that she needs the 2nd and 1st skills raised up for that. Emiya can wait until all his strengthenings are out

Chloe feels most balanced with no weakness. St np, evade, self buffs..

I assume your servant are decent leveled and you're a high level master. As people mentioned above, their ability have been explained.
Mostly all of your Gold Archer is an Art type. Emiya, Chloe, and Orion is also very strong Boss killer.
As a boss killer they also have different uses based on the situation.
Gonna add a little, it's up to you to decide.

-Orion : tough solid Boss killer with main Niche of slaying males. Outside of slaying males 'he's' still good to other uses.

-Chloe : ST archer Art Crit NP spammer and bonus hit real hard with her NP lvl 5. Use her only if you don't like such long fight. Pretty much your general Boss killer.

-Emiya : He's tougher than Orion with def up and spammable attack down. Unless you don't like balanced team and go through with brute force don't use him.

Well, FGO full name truly is 'Fate Grinding Order' with farm as the biggest of it's content.
Wisely i recommended Emiya, he can slay boss, daily farm with his AOE and also good for dealing with high hp mooks.