What is the easiest way to "listen" to the network? If I want to automatically check up on payments and ensure they are not double spends and check the confirmation count, what is the easiest way to do so?
https://blockchain.info/ , 'nuff said.

They feature live events of transactions appearing in the network, as well as blocks.

Other than that, there is this - http://bitcoinmonitor.com/ . It shows such events graphically, along with currency trades.