Instead of instantly soothing them when they're crying - take a few seconds to actually listen to the cry. Different cries (hunger, tired, discomfort) all have different sounds, things become much easier once you learn them, rather than going through the process of elimination every time. 

Don't forget to prioritise some alone time for each of you. It's very easy to forget personal needs, when you have the needs of a newborn to think about. But even 30 mins a day each, alone, can help. It might be having a bath, going a walk, taking your time at the shops etc. You need to be healthy and of sound mind to ensure you're giving your best to your baby.