Determine the rules in your local jurisdiction to see if you need a permit and what you need to do to apply for the permit. Most likely you will need to get one for building and electrical. Follow their instructions on when to call inspectors.
Design what you want to do including details.
Remove anything that you do not want in the finished room.
You will need to put a footing where the garage door currently is. Start by removing the concrete across the length of the door and about 3 feet wide. Dig out an area that is big enough to put a footing in that meets local building codes.
Put rebar in the hole. Your building inspector will make sure you do this properly. Build a form for the concrete and coat the inside with used motor oil. Get an inspection.
Pour concrete in footing. Make sure to vibrate the concrete in the form to remove air bubbles.
The concrete should be set in about 3 days and you can remove the form. Remove any excess concrete.
Remove existing garage door.
Frame in wall. The bottom plate should be treated. Generally this wall should be made with 2X6's, 16" on center.
Backfill dirt around the footing.
Put exterior sheething up using 1/2" CDX plywood or OSB. Cover sheething with a vapor barrier.
Add siding to match existing siding.
Now you have a secure space and it is protected from the elements.