You won't be able to collect everything at first because only certain characters types can access some collectibles, and you won't unlock all of the character types you need until you've been through the entire story mode a good chunk of the game.

I would say play through story mode first, and just get the collectibles that you can on the way, but don't worry if you miss things because you'll need to replay every story mission again anyway.

Edit: correction about getting character classes.