If you have large trees around your home, no doubt you have encountered the challenges of growing turfgrasses in the shade. Most of our southern warm-season turfgrasses, such as centipedegrass, need 6 to 8 hours of full sun to perform well. At the same time, the shade trees are intercepting the light, their roots are competing with the grass for moisture and nutrients. So, what I usually suggest is that people keep the two areas separate. It is best to maintain a beautiful, healthy lawn in the sunny part of the yard, even if it is only a small area. Moss typically grows in moist, shady areas with compacted and acidic soils. Therefore, in these shady sites with trees, cover the ground with mulch (such as pine straw, hardwood mulch or pine bark nuggets), out to the tree dripline (that is, where the branches and leaves stop). There are many species of evergreen groundcover plants that grow well in the shade, such as monkeygrass (a.k.a. bordergrass or liriope), mondograss, and various hardy ferns. For more information, see HGIC 1100, Groundcovers: Planting and Care and HGIC 1176, Hardy Ferns.
Moss is a naturally occurring groundcover that people seem to either love or hate. I would group myself with the moss lovers, because I like its soft texture and bright green color. Many people, though, would rather have a lawn instead and think the moss is “taking over” the lawn. My suggestion would be to either mulch over the moss, plant a groundcover, or change the soil and light conditions to make it more conducive for better turfgrass growth rather than for the moss. The first thing to do, as usual, is to have the soil tested, so that the acidity can be corrected by a lime application according to the recommendations for YOUR soil. Next, relieve the soil compaction by core aeration. Additionally, remove some lower tree branches (but NOT by topping) to allow more light to reach the ground. For more information on controlling moss (as well as algae) in the home lawn, please see HGIC 2363, Moss and Algae Control in Lawns.
If you go with the mulching option, consider whether people will be walking through that area. If so, use hardwood or bark mulch, and maybe add some stepping stones to create a clear pathway. Never cut off surface roots of trees, but instead, place the stones and/or mulch between them. I like pine straw, but it tends to be a little slippery, so only use it in areas with little foot traffic.