Who should I level skills up?

It depends a lot on your needs. But I would vote for Drake. 5-star servant and good for farming.

Mash.

Chloe to 3rd skill to 10 can weaponize her arts cards into literally vomiting stars for 3 turns, thank to her 6-hits hitcount. Plus the reduced cooldown is very nice. Upgrade the damage steroid to at least 6.

Assassin Scat is a farming machine, only the 3rd skill deserves proper materials to be wasted on, upgrade it to 6 and call it a day.
Upgrade Magecraft of Kiritsugu to get a stronger NP and a better refund from it.

Caster Liz Innocent Monster is the strongest passive continuous stargen of the game, give it lv 10 for 12 stars for 3 turns on a cooldown of 5. Pair her with an Archer/Rider/any servant with a CE who gives him a star absorption and all shall be good. Even better if you have a Hans, free stars raining.

Rider Drake and Rider Kintoki are monsters, all their skills are worthy of maxing (maybe not the cure of Kintoki, still, at least lv 4 is nice).

I recommend to leveling Chloe / Kintoki, because theyre NP5 right?, i would pick NP5 4-star instead NP1 5-star servant.

Mash. She is basically always valuable - and it's quite easy to level up her skills.

After her, probably Drake - her self-NP charge, NP damage boost, and AoE Noble Phantasm make her quite handy overall for the grinding where you'll spend most of your time (and even a fair few bosses).

Pick for Mashu, Caster supp and Farmers like Drake.

You will use Support for almost all fight and using Drake a lot for AQE clearer and QP farm is really needed.

Sry if i'm not wrong where's your Hans? You have bunch of Caster but a lot of them is DPS and healer. Considering material for raising Hans is fairly cheap, also he's so tremendously can help a lot of your team comps.

You also have bunch of Buster oriented servant. Raising Shakespeare is gonna be a big help. Also he's cheap to raise.
Marie Summer is also good choice for Offensive Crit style team comps.

That's it, the choice is yours :)

Are you dead set on just picking one servant and maxxing all of their skills? Not a lot of servants have all 3 skills really worth maxxing. At least most servants have a skill or two that isn't necessarily a priority. It might be more lore/QP efficient to pick specific skills that are higher priority, and max those first, rather than maxxing all 3 of one servant's skills. But that's just my opinion. As for which ones you should pick, it really depends on which of those servants you prefer to use. I.e. if you use Chloe a lot, then definitely max her 3rd skill. But if you dont use Chloe, then it would obviously be a waste to max any of her skills.