You will not be able to madk the sound, so consider the most dound proof windows for the front. If headlights bother you, a screen of shrubs can help there. I would focus your effirts on msking the rear yard a sanctuary from the road. FWIW I have lived on a busy rural road like yours and became attached to it. It was nice to have a feeling that my gardening efforts had an audience and I liked having people slow down and wave or chat. If you have children, perfect gor a kemonade stand!

What is the gray out front?

Cement pad. I believe there was a porch there before that the previous owner removed

I would get rid of the cement pad...very unattractive and it takes up half of the front yard. Then I'd lay out a shrub border as close to the road as you are allowed. Where are you? Are snow plows a problem?

NYS. We have snow plows.
We had someone suggest evening it out and putting a trellis to a slat wall facing the road and leaving room for a shrub border.

I lived in a riverfront condo close to the road. Totally worth it.

If I had children or pets, I would constantly be worried about the road. I've lived on a rural road and the traffic really bothered me, but where I lived there were logging trucks going by. Very loud and they shook the house as they passed.

Where is the property line for the right of way for the road? Can you put up a fence?

Looks like every single tree that used to do exactly what you want were just cut down to sell the house. And they didn’t grind the stumps so now anything you’d plant will struggle due to all the roots that are still alive. Ugh. Maybe those trees were dying and needed to come down. Who knows.

Anything you do will take some good landscaping ideas. And possibly understanding the possibility of rain runoff that you’ll have instead of those trees to suck up all that water.

If stumps or roots still send up suckers, you clip them until the tree exhausts itself. A dead stump is more like a nurse log for other plants. If you want turf, you will want to get rid of them but no big desl.

The most obvious challenge is how do you or your guests drive up to the front door… or park …to use the front entry?..a half circle of solid fencing….surrounding the door and two side windows big enough to create a walkway and an attractive garden entry….and to provide a good sound barrier….

but it would require a driveway and several flat parking spaces…and may even need steps from the parking area ….up to the entry area…

You haven’t got a picture of the rear of the house…but if the interior layout would allow a rear entry ..you could include a deck with steps for a rear entry

I have a somewhat similar situation with my new house quite close to the road, though I'm actually on a street in a very, very small town. Large shrubs, set back as required by road easement and to avoid salt is what you want, some evergreen if they will grow in your zone. (My evergreens are Viburnum chenaultii and V. pragense.) If you can, remove the cement. I have a 4 ft wide path--crusher run--along the front of the house to the front door from the driveway with the shrubs on the street side and low ground covers near the house with just a little height provided by 5 tall narrow grasses on one side of the door in the middle of the ground covers and a tall native in the bed on the other side of the door.

All was planted this spring, so it'll take a few years.

I think lacyvail has the right idea. A senicircular drive for parking and drop offs plus a mixed shrub border along the road. Since you’re in NY you probably have deer, which will narrow your shrub choices considerably. Don’t waste a penny on anything thats not on that short list!

A mixed and staggered border along the road should blend into the rest of the roadside plantings. Anything very different in formality (like a uniform row of pointy conifers) will be jarring and make your property seem small - unless you have very wide street frontage, in which case it would break the bank anyway!

You could look into a stone fence or better yet, a gabion wall — excellent sound control because of the irregular shape of the rocks interfering with the sound waves. Multiple Stories on Houzz. Here’s one: https://www.houzz.com/magazine/garden-walls-gabion-evolves-from-functional-to-fabulous-stsetivw-vs~44603134