Which Askr trio member to build?

Build them all.

Alfonse, unless you don't feel you can get him done in the next few days, as he's currently a bonus ally, and will be again next week. So you can get some real mileage out of building him right away.

I've only built Sharena so far and she is currently one of my favourite units I've built.

Alfonse is the cheapest to build effectively and probably the strongest of the three. Refine his personal weapon for res. Make sure it's his res weapon and not his special refine. Defense can also work but I suggest res. You can build him with Brazen Atk/Def from Ares or a Brazen Def/Res from Kana to stack with his weapon. For a special, glimmer if you need 2 cooldown, bonfire for 3, and Aether for arena. A good B slot for his would be either QR3 for pre-brazen damage or vantage for ultimate cheese. guard also works just for surviving. I suggest vantage as you can get it the cheapest from the best character in the game, Gordin, at 4 star. C skill default would be Atk Smoke from 4 star Kaze, or any buff you may need. seal, Brazen Atk/Res.

Next would be Sharena, don't worry about that yet though just finish one.