Step one is the inspection, and it should happen before you call your insurer. A roofer who gets on the roof, photographs the damage, and counts the hits per square gives you the facts. If there isn't enough damage to justify a claim, you've saved yourself a claim on your record for nothing.
Step two is filing. You call your carrier, give them the storm date, and they assign an adjuster. Ask your roofer to be there for the adjuster meeting. The adjuster is on dozens of roofs a week and can miss things; a contractor who has already documented the damage makes sure the scope covers the whole roof, the gutters, the screens, and anything else the storm took.
Step three is the scope and the money. The insurer sends a scope of work and an initial payment, usually the replacement cost minus depreciation and your deductible. The depreciation is released after the work is done. A good roofer builds the job to that scope and supplements the claim if hidden damage, like rotten decking, turns up during tear-off.
Step four is completion paperwork. Once the roof is on, the roofer sends a certificate of completion and final invoice to the carrier, which releases the recoverable depreciation. You pay the deductible; the insurance covers the rest. From the first inspection to that last check, you should be dealing with one contractor, not three.