I just successfully received a claim from Fedex this week. The keys in my opinion are:
1)ABOVE ALL, carefully inspect the package before you open it. Take photos of it before, ,and during unpacking. Take photos of the damage. If the driver is still there and you notice damage to the package - tell him to note it in his log. And you should make a note next to your signature.
2)Keep all of the packaging intact as much as possible. You'll need to show it to them.
3)Normally they only deal with the shipper. If you're the receiver you may want to manage the claim and will need the shipper to send you a letter of authorization to handle and receive the claim payment.
4) Fill out their forms clearly and completely. You'll need a bill of sale, and repair quotes documented. Follow up by phone to make sure they received them and they have everything they need (I had to re-fax some info twice). Be a pest, though polite
5) Evidence that the damage was caused by their improper handling needs to be compelling. Spell it out for them. That is, you need to be able to show it was packaged properly, and that there is evidence of mishandling.
6) A claims processor will call you and tell you to re-package the item in it's original package and they will have someone pick it up for inspection. Insist they send an inspector to your home so you can show them the damage. Explain it's fragile and expensive electronics. If you give it to them, you may not see it or your money again. UPS did this one time for an amp that had a bent corner on the face plate. The denied the claim and returned the amp to me with a matching bent corner on the other side.
Keep following up to find out what the status it.