Trees take time, but some grow faster than others. You might want to do some research (and the Trees forum is a helpful place) about what trees - in your area (so tell where you are)- would grow relatively fast. Mix in a few shrubs too for interest. Nature does it that way.

Surprisingly, tree seedlings can grow very fast, sometimes catching up to purchased larger trees in just a few years. In Georgia, you can purchase tree seedlings very economically in 50 unit bundles from the Forestry Commission. Pine trees are the cheapest of all and can be used while the hardwoods get bigger (then you can remove some of the pines if you want).

I think it's a great idea ... I've been trying to convince my in-laws to likewise get rid of the several acres of grass that they mow for no good reason except that it grows!

Here is a link that might be useful:

Letting the grass go will result in a yard of grass and weeds for several years, then several years of weedy shrubs, then some spindly trees. If you don't mind the weedy look you can go that route. Much better to pick a small area then kill the grass, cover with mulch and plant trees. For faster woods plant bigger trees. Every year you can increase the size of the area. Planting native trees like cherry laurel is a great idea, research what trees are native to your area (see the link below for south eastern species). Cherry laurel is more shrub than tree, consider some big trees like hickory, oak and pine to create a real woods.

Some counties and conservation authorities will provide seedling trees at very low cost to land owners who will plant them. They might also provide seminars on tree selection and care.

Here is a link that might be useful: Native trees of Georgia

Get your hands on this book 'Gaias Garden' by Toby Hemenway. k.c.

Waiting for woods to take over the lawn could be a long wait. I would get whatever trees you like that are native to your area and plant them in the appropriate locations. You could put down big tarps, cardboard, etc. to suppress grass and/or mulch as suggested above. You could also just selectively kill three or four foot diameter circles of grass and plant your saplings there. You do not want your saplings to have to compete too much with the grass.

Letting it just 'run wild' can allow annoying stuff to grow, such as wild grapes, honeysuckle, poison ivy, kudzu, just to name a few. With a little effort and some tree seedlings, you can grow a forest. Try to avoid trees known for weakness, such as Bradford pears and silver maple. It's good to plant a mix, so if a disease should show up, you won't lose everything. Good native trees (for your area) are always best, and you can plant fast growers for instant gratification, as well as slower growing legacy trees for generations to come.

Your best approach would be to purchase saplings from your county extension or forest service/DNR.

I'm getting 150 trees pretty soon here from my Soil and Water Conservation District office. Saplings of about 2-3' are $1.25 each and seedlings are $.65 each. To put this into perspective, the bur oak saplings are 2 feet tall, and are already 3 years old when I get them.

Waiting for the forest to encroach on a field could take a very long time (your lifetime). We stopped mowing portions of our land 3 years ago, and while we see some new stuff moving in, it's slow. Right now, the only volunteers we have coming up in the yard are a few wild plum, lots of choke cherry in a small area, and multitudes of rugosa rose bushes among the pines that were planted 11 years ago (and are now 15 feet tall!).

Planting trees to forest your property will take a significant time/maintenance commitment. You have to make sure the new planted trees get enough water, have the nutrients they need, and have protection from extreme heat/cold and from hungry little critters.

I'd recommend starting on a smaller scale planting different sized trees and different kinds that have different growth rates. Make sure you mulch around the trees at a depth of 2-4 inches in a 1.5 foot radius around the tree, but not touching the mulch to the tree. This is especially important if you can't water the trees and it gets hot/dry. The mulch keeps the soil cooler and retains moisture.

Better to buy and plant trees that you know will work than to let the woods take over: most invasives love to take over unmowed lawns and clear areas.

Oaks and hickories are always good choices - good luck!

Research "Pit and Mounds"

Where you live, and not just your zone, would be a big help.
I'm Zone 8 also, but I'll bet my climate is a lot different than yours as far as recommending trees goes.
Native evergreens for those areas where privacy is needed at the edge of your property should be a primary concern.
Deciduous trees with evergreen shrubs in front of that. Pretty soon it will all fall in place. That's what I did.