… or can you pick up multiple Goals as you’re moving?

A: You can claim a Goal card at any point during your turn. As soon as you meet it, you get it! So yes, you can collect more than one in a single turn.

Instead of thinking of it as a “move phase” or picking up Goals “as you’re moving” remember that each “step” you take is discrete (separate), so you can move one space, play a card, then take another couple of “steps” if you have moves left (as noted in the paragraph under Turn Sequence, in the rules).