I'd say your first action should (have been) be to photograph the bag in situ in the baggage hall. If there is identifiable surroundings it can help to prove the state you received it in. One of the advantages of modern mobile phones and turn on the 'date stamp' function if available.
I've only had to claim for badly damaged bag once (arriving LHR as it happens) and the carrier paid up right away. I think that prompt action is always best.