Should my toes be completely curled up in my climbing shoes?

The toes must have close contact with the front of the climbing shoes without being pinched so hard it hurts. It's a balance between getting the best contact between toe tip / shoe / climbing wall without ruining the climbing experience because the climbing shoes hurt.

Try to push the toes all the way into the shoes you wear on a daily basis so that all toes have close contact with the front og the shoe, and then imagine that the toe tips should grip the sole at the same time. That way the toes automatically curl a bit, without hurting. That's the feeling you should have when climbing.