I just purchased the SM58 Vocal Artist Diversity system, and though I'm quite happy with it, I do find it irritating that everytime I breath into it just a little, it can be clearly heard through the output speakers, very loud and with a lot of bass. Also, I find the mic to have an extreme 'bassy' sound (It's only clear when I hold the mic very far away from my mouth). Am I doing something wrong, could it be the connections or is this just this type of mic?
You are describing "proximity effect"; a characteristic of unidirectional microphones like the SM58.
Microphones cannot tell the difference between "breath noise", "speech", or "singing". If you move air into the microphone for any reason, the microphone will reproduce it.