Print a flat map of the Earth in black and white. Cut out the continents carefully.
Glue the cutout continents to one of the Styrofoam balls. Apply glue to the backside of the cutouts and press gently against the Styrofoam. Don't worry if there are wrinkles. The wrinkles will look like mountain ridges when you paint them later. Find a 3D view of Earth to help you know where to place the continent cutouts.
Create land mass and oceans. Paint the continent cutouts using brown acrylic craft paint. Let the Styrofoam ball dry for one hour. Apply green acrylic paint to some areas of the continents to create fertile green soils. Avoid painting Antarctica green because this continent is any icy continent. Paint the rest of the Styrofoam ball with blue acrylic craft paint to make the oceans.
Paint the entire second Styrofoam ball with yellow acrylic paint. Set aside to dry for an hour.
Pour some orange acrylic paint into a plate. Dip a sponge in the paint and dap it all over the Styrofoam globe to create swirls and streaks. These will be the solar flares.
Sprinkle plenty of orange glitter all over the Sun model while the paint is still wet. This will create shiny Sun spots.