I wont comment on the types of lawn seed ---they would be ideal for southern lawns and I yield to the manufacturers.

First off...congratulations on owning your home. Many years of happiness is hopefully in your future.

But....that saidt, home ownership and obtaining a nice lawn can be very irritating on a new gardener---one does not necessarily follow the other.
Hence the reason you are in this journal.

You have many trees you say. Then I suggest you do some research into why trees and lawns do not get along well.. They are mortal enemies.

Trees have roots that thrive on taking water from the ground. They take what they want and leave little for other plants.
You can help your ground keep as much grass under the tree by letting in as much sunlight as possible. Cutting lower branches can help. Removing some of the canopy of deciduous trees can also help....but don't ruin the look just to let in a few more rays.
Feed your lawn regularly.

That does not necessarily speak of commercial fertilizer.
Organic matter you can inject into the soil of the lawn will certainly help retain moisture and aids the structure of the soil to help the roots take in nutrition.
It can be an expensive way to feed your lawn but, if you can afford it, once you have your lawn coming along, apply 1/2" - 1" of topsoil.compost/triple mix over the area every spring whether you believe it needs it or not.
That one thing, can bring about you having a great lawn---one that can put up a fight against the insects that will invade it.....and they WILL come.
Let the grass grow high...3" - 3 1/2"...do not even consider cutting it lower unless your lawn type can handle it.
High grass shades out weed seeds and while you may have to mow it more often, the benefits outweigh the labor.

You have a dog. That is another enemy of lawns.
Better get use to seeing bare patches and areas that go brown. Dog urine is one of the most acidic materials that act against grass roots. There is absolutely nothing you can do to change that fact.
Do not even consider changing the pet's diet to try to change this ---it cant happen. All you will do is make your pet sick.
Water is usually only the one thing that can affect how bad the acid damage is.
Get use to cutting plugs from areas of your lawn to replace the damaged areas.

Water your lawn according to its needs.
Do not water on a schedule....only as it needs it.
If, by chance, you are on a softening system for your water, do not use this on yoir lawn....it would be cost prohibitive anyway. Use a tap off the system.

Now because I've said all this--
welcome to home ownership.

If you applied an inch of soil every spring, in 40 years you would have a yard 40 inches higher than it is today. How much sense does that make? Even 1/8 of an inch every spring would raise it 5 inches in 40 years. Please don't bring in any topsoil, ever, unless you have a low spot to fill in. Here is a picture of topsoil gone bad where they ended up installing a landscape barrier to contain the growing height of their lawn.

{{gwi:93555}}

What about compost? If you applied 1 inch of compost every spring, your grass would spend the rest of the season trying to grow out from under a pile of compost. I have neighbors who apply 1/2 inch every other spring. It takes their yard a full two years to grow out from the smothering effect of too much product. Compost has the feature that it does not accumulate over time like topsoil, so if you used compost, at the end of 40 years you would have the same soil height as you have now. Compost has the problem that it costs a fortune. Even at the recommended rate of 1/4 inch, the cost in my neighborhood works out to be $70 per 1,000 square feet after delivery charges. On the other hand I can fertilize with a really good organic fertilizer for well under $10 per 1,000 square feet.

Yes, dogs are one of the enemies of lawns. But only because they tear it up, not because of their urine. Normal urine has no acidity. Unless the animal is sick, the urine has a pH right around 7 (normal). What dog urine does have is urea, a reasonably strong fertilizer. But since you didn't mention dog spots, I'll save the rest of the tutorial on that for someone who wants it.

You have shade, kids, and dogs. If you did not have kids and dogs you could have a lawn. If you did not have shade, you might be able to have a lawn. But you have the triple combo of death. Why? Some grasses have the ability to recover from wear and tear. Unfortunately those grasses only grow in full sun. The grasses which can grow in the shade will die fast with kids and dogs.

You might consider a several inches of tree mulch. The arborists you see hauling around a chipper/shredder behind their boxy looking trucks are the ones to talk to. Like compost, this stuff will simply evaporate over the years. If the dogs slow down and the kids start to use the yard less, then you can come in with a fine fescue type of grass and have a lawn again.