For each spell level you have access to, choose one spell of that level to be a signature spell.
Does this mean at 3rd when you gain the feature you gain both a 1st and 2nd level signature spell? Or are signature spells behind a level or something?
Well at 3rd character level when you get Signature Spells, you gain 2nd level spells (I know -.-).
But yes, I think it's quite clear you pick a 1st level spell as well.
"For each spell level you have access to". There is no ambiguity there. You have access to 1st level spells, you pick a spell of that level.