Gilgamesh or Jalter?

Here's my opinion as someone who has both Gilgamesh and Jeanne Alter.

As a new player who just started out your best bet is Gilgamesh. You simply don't have what you need to make Jeanne Alter the Buster Crit monster she's meant to be as a brand new player. Gilgamesh is a very self-contained servant. He has high crit star generation, the hit counts to use it, high crit absorption, can boost his own NP generation gain, and eventually gains a NP self-charge ability.

On top of that in addition to his upcoming rate up he'll be getting the interlude that makes Enuma Elish truly monstrous. It may be an AoE NP but once he has that Ea is an AoE NP with the power of a single target NP against the overwhelming majority of servants. He's a bit fragile himself, but Mash can help him with that.

Now, while Jeanne Alter can put Gilgamesh to shame in damage output the problem there is that she absolutely needs her team to be built around her to do so, needs crit stars, and is a much more resource intensive investment than Gilgamesh for someone that is new. She also generally needs someone to help compensate for her fragility more than Gilgamesh, and in the moments she doesn't have crit stars she performs similarly to him in a situation where class advantage has not come into play.

Roll on Gilgamesh. You can't go wrong with him and Mash is enough to give him the support he needs to survive. Jeanne Alter will be available later this year. If you find yourself with some good crit generators and support later this year then take a swing for her.

Go for Gil, but if you ever get him, try to keep raising your Robin.
Reason to go for Gil is already stated by Delnoir and the reason why you should raise Robin is because he is a very strong archer with disgusting damage potential, and he doesn't require a complicated team comp with 5 stars exclusive servants to be able to deal 100k-ish damage regularly, unlike Jalter. Also, you will need multiple servants in the same class, so, don't worry too much about him being in the same class as Gil.

So let's talk about the game then both characters. This game requires almost no servants from the gacha to beat the game. I'm pretty sure you could play with only 3*'s and beat the game, it might be difficult at times but no servant is required to beat the game (except probably Mash, but you already have her).

Now onto Gilgamesh (Gil) and Jalter. Gil is pretty self reliant. He can generate his own crit stars, collect them, and boost his attack, All and all awesome guy. All of his attacks are multi hit so he can generate NP as well. He has no defensive skills so you will have to support him in that department (looks at massive shield Kouhai). His NP will be discussed later.

Jalter currently (and probably forever) has the highest Attack of any servant. She does damage and more damage. She can collect and generate stars (not as well as Gil) and can boost attack like Gil. She also comes with a self invincible skill (which will be glorious to dodge NPs) but cost her Some HP to use. If you need damage she can do it.

Now their NPs. Gil is a single hit AOE which is Buster (so he can Buster Brave Chain it for lots of damage) but that's about it. Jalter is also Buster but is single target. She can also Buster brave chain and her's strips Buffs and curses (poison essentially).

Now after seeing all that you need to decide is what do you want to prioritize. AOE wave clear or Single target delete button.

(Now for my own BIASED opinion)
Gil is an awesome servant, I loved him in the anime and threw a lot of quartz at him because let's be honest who doesn't like throwing swords, spears, axes, and god knows what from a magical portal at people. It's freaking badass. BUT aside from his badassery his NP does a single hit of damage. And because of that he is a meh in my book.

Jalter does damage (cool I have said that enough by now) which is awesome. And the single target delete button is pretty awesome as well. I prioritize single target because those bosses with 200k HP aren't going down to an AOE one hit NP.

Okay so now back to Robin vs Gil. Both do pretty different things and would work well together.

I don't think I made this clear earlier, but pick servants you like to see and generally love. One saying around the reddit page is "Grail for love not gameplay" I love Jekyll but he really doesn't do his job well (as an assassin or beserker) but he sits in my party and I would pick a 3* craft essence over a 5*one because he is on it( I am pretty blind when it comes to Jekyll) so if you love robin, well then keep him and grail him up to 90!!! (Again I'm not an expert on love nor Robins uses, so take my advice with a grain of salt)

Back to Robin vs Gil. Robin does huge damage by poisoning then NP to a single target and Gil does AOE damage. Robin might get neutered if people resist debuffs, Gil won't, but then again AOE single hit of damage. Robin can provide a little more support with his Attack down and poison, and can save himself with his evasion skill. You can keep both in your party if needed (like where all the enemies are sabers) or switch them out while you farm up Gil. IIRC Gil and Scathach usually come out at the same time in gacha and it never hurts to go for Scathach (super biased here and I love her, I originally was pulling for Gil and not her, got her and am now a true believer in her power)

Sorry for the stupidly long post, best of luck pulling.

TL:DR Jalter = single target delete damage, Gil = AOE damage, follow your heart, multiple archers is never a bad thing.

I can't add much to what was already said, but I will just say a few short things.
Jeanne Alter needs supports to really shine, Gilgamesh does fine on his own. Both are good, just for different team comps and situations. There is no wrong decision when you choose between them.
As for Robin, it never hurts to have more than one strong servant of a class. Especially Archer. I heard the next chapter, Camelot will be full of Sabers. You will be glad to have multiple Archers.