The first thing you want to do to repair roof damage from storms is determine what the damage is. Taking a quick look over you can determine whether or not you will need to buy new support beams, or if all you have to do is replace a couple of loosened shingles. If you have an attic it is a definite plus to go up there and check out if there was any structural damage to the wood, often there won’t be, but in the case of a severe storm with hail and flying debris there might be an underlying amount of damage.
With the amount of damage accessed you can determine if whether you need to buy some structural reinforcement beams or if you just need to buy a pack of shingles. So now that you have an idea of the repair requirements you can either hire a professional to fix the damage, or if it is slight enough, and you are knowledgeable in the repair field you can gather the materials to fix it yourself.
If you are fixing the damage yourself you most likely will need to purchase a pack of shingles and have a roofing nail gun at your disposal. Roofing nail guns work on the premise of air pressure and are connected to a engine that will build up the air pressure so you can sufficiently drive the nails into the shingle securing it in its place. So by using a nail gun climb up to your roof and begin tearing down and replacing torn off shingles. This of course is done if there was no damage done to the underlying beams, if there was you will need to look into reinforcing those then start the reshingling process. After nailing in shingles to cover up any holes and replace the old ones you can go back into your attic and verify that the shingles are in the right place and secure. As far as testing the roof goes you can run a water hose on top of it to ensure all the shingles were put in place.
So overall the process of repairing roof damage is similar to the original roof building process, just the verification of stable support beams followed by replacing shingles.