Moss growing in a lawn is usually the result of one or two of several causes. Whatever the specific reason(s), the moss grew there because bare ground was available and grass was not competing for the same area. The challenge is to find out why grass was not growing in the bare spots on your lawn.
Check the pH of the soil with your testing kit, preferrably one that will also test for nitrogen, phosphorus and potassium. Follow testing instructions, which usually call for taking a soil sample, mixing it with the powder provided, and then with water. Stir well and compare the resulting color with the chart included in the kit. The results will tell you if the soil needs lime or fertilizer.
Remove as much of the moss as possible either by using a chemical such as ammonium sulfate or copper sulfate, or by manually raking up the clumps of moss. Both of these will only temporarily remove the moss if the conditions that allowed it to grow in the first place are not corrected.
Apply the necessary amendments and work them into the soil with a garden rake. Try to scratch the dirt about 1/4 inch deep. If the soil if very compacted, you can start aerating it naturally by adding compost, which will in turn attract earthworms. Punch it into the soil with the tines of a pitchfork.
Improve the light reaching the lawn by removing as many limbs as possible that are blocking the sunlight. Sometimes, just thinning the overhead branches will allow enough sunlight to get through. Call an arborist to remove very large limbs if you do not have the proper tools and training.
Reseed the area by sprinkling the new seed over the bare areas. This is much faster than waiting for the existing lawn to fill in, although it should once the grass growing conditions improve. Add about 1/4 inch of topsoil/potting soil or sand over the area to make sure the grass seed has good soil contact.
Water the area daily until you see the grass seed sprouting and then only once a week. Keep all traffic off the new lawn for about a month to give it a chance to establish a good root system.