My heroes!! It's so nice to hear that the word about invasive plants is getting out, and your desire to rid your wild area of Japanese Privet (Ligustrum japonicum) significant threat and Japanese Honeysuckle (Lonicera japonicum) is heartwarming. It just so happens that the same two invasive plants pose significant threats to natural areas near my Central Texas home. We have whole workdays devoted to Privet, Honeysuckle (and Nandina) removal!
But to answer your question, you will find that upon removal of the light-stealing villains, many understory trees and shrubs will thrive in your native woodland. Bring in mid-sized trees that offer beauty in all seasons, such as Redbuds, Buckeyes, native Plums, and Sumacs. Any number of shrubs will offer berries, fruits and cover for many wildlife species that call these woods their home. As you come to the "edge" of your woods, where it meets the more formal part of your yard, gradually step the plants down in size to create a transitional area of blooming trees and shrubs, plants that have fall color, bunch native grasses, and perennials. In effect you will be sealing the "edge" of the woodland beyond, allowing it to remain the shady habitat it was intended to be. All too often we open up the woodland edge so we can look inside, but we do this at great cost. Common in developments with adjacent lots of fewer than five acres, healthy woodland can be compromised when homesites are carved out, creating extensive "edge." Many folks find that this sort of buffer is all the garden they need since it looks beautiful all year and it offers much to wildlife. If you have a deck at the back of your house, you could have a walkway to a more rustic patio near the buffer. I haven't seen your yard, but chances are that you may not need any lawn area at all.