Wash the floor. Use a pressure washer to completely clean the garage floor. All dirt and debris should be removed.
Insulate the walls. If the walls have been framed in but have no insulation, choose a grade appropriate for your area and roll this out in between the frames. Nail it into place.
Install wall panels. You can use pegboard panels to create shelving place and a spot to hang toys or you can drywall in the garage, prime it and paint it. Chalkboard paint on one wall will give kids a place to practice their art.
Cover the floor. Paint the floor with epoxy flooring paint. Keep the doors open while you do this for proper ventilation. Use a floor paint applicator to make the job go quicker. This will seal in the floor.
Apply vinyl floor mats to the floor. Soft vinyl garage mats that piece together will give you an outdoor friendly, but kid-safe flooring option. Alternatively, you can install a carpet pad and indoor/outdoor carpeting, but this will be somewhat difficult to keep clean.
Install lighting into the space. Unless you already have plenty of lighting in the garage, it may be quite dim. Have an electrician wire in a few overhead lights into the roof joists and a switch. Overhead lights are best, since they will be out of reach of small children.