You can do either. If you have a couple grand available, I would probably pay for this relatively minor claim out of pocket. If you have another claim in the next couple of years, you would get dropped and end up with a high risk carrier. Their products are not only more expensive, but they are less comprehensive.
What you should do immediately is contact a restoration company and/or a contractor you know. You should get someone in there with fans to dry out whatever can be dried and remove drywall that cannot be dried. I am not endorsing any of them (all of the ones I know are solid, I just don't endorse anything on YA!), just look for the ones in your area that can respond TODAY. The faster you get them in, the less damage you will have.
Insurance, assuming that your coverage is standard and valid, would pay for the water damage but you will need to pay to the damage to the roof.
I would contact the builder/contractor who installed this roof along with the manufacturer of the roof material. 3 years old and you are having these problems, sounds like a material defect or negligent installation.
The rule of thumb is if the repairs will cost more than 2 times your deductible, you should have your homeowners insurance pay for it. However - your homeowners insurance premiums will go up, possibly for a number of years, depending on the amount of the claim. That's why it's not worth it to file a homeowners insurance claim if the repair is small - you'll end up paying for your deductible, plus you will pay increased premiums.
If at all possible, if you have a knowledgeable friend or relative who can look at the damage first before getting an estimate, you can avoid getting ripped off by a roofer as well. They will try to sell you a tear-off, which may or may not be needed. However, if the roof really is 3 years old, you probably just have a little wind damage and shingles are lifted up and letting rain in from a certain angle.