To add, I was thinking about placing 2 of the red oaks along the back property line (in between what will later be groupings of conifers). We plan to put a few conifers in each back corner and a 3rd grouping in the middle of the back property line, so along the back would be conifers, oak, conifers, oak, conifers, from corner to corner. Then, I was thinking I would put another red oak on the right/East side property line toward the back right corner (but not all the way in the corner since a grouping of conifers will be in the actual corner) and then the the autumn blaze maple also along the right/East property line, but a little closer to the house ( though still at least 25' away). This is not symmetrical, but it would keep the deciduous trees away from the patio. We would then likely balance with a row of arborvitaes along the patio in the left/West side to give some foliage and privacy over there as well. Thoughts?

Here is an example rendering of what I was thinking above. The red trees are the oaks and maple. The round green are the conifers. This is not my actual home, but a similar home in my neighborhood. Our yard is actually deeper than this one by a good 10-15'. Also, our patio is larger and comes out more (probably another 50% more) than the patio/deck in the photo here.

I would not place plants as you propose. The alternating scheme lacks the appearance of sophistication and tends to look "busy." Where unlike plants meet, the one quickest to achieve height will dominate and "steal" the light from the shorter. While it probably won't be to the point of total eradication, a tree, for example, will decimate the lower foliage of a nearby growing evergreen. For that reason, it's best to minimize where this would happen and try to create some space between these two types of plants.

Not sure what conifers you intend to use, but it's likely for many years that the trees will be taller. It would look better if these taller plants were near/at the corners of the yard rather than at the central portions of the property line.

Insofar as trees needing to be a certain minimum distance from one another, it depends entirely on your purpose. Are you going to be growing them for resale in years hence and need all sides of their individual canopies to look good? Probably not. Therefore, it may be preferable and more useful that you combine trees into a tightly clustered group and create a larger, more substantial canopy held aloft by three trunks. The interior of the collective canopy would never been seen. The greater number of trunks would make a more powerful artistic statement as the trees age and the canopy is limbed up. The picture shows a simple, and I think a more useful way to arrange these two elements. Of course, you'd need more than this to make the yard complete.

Yaardvark - thanks so much for your feedback!

To answer your questions, the goal of all of these trees, both conifer and deciduous, is first, to create privacy, and second, to give the yard visual interest. Because the home is a new build, there are no trees and we currently stare at the back of our neighbors homes. I love when I visit a friend in an older neighborhood and the yard is encircled in large mature trees so I am trying to put trees in now that will give that look to our yard long term. We plan to be in this house for a very long time.

The conifers that we put in (hopefully in a few months in the Fall) will only be 5-6' tall because we cannot afford anything larger and they will probably be a mix of white pine and Norway spruce, again, because they are cost effective. The red oaks and red maple are very small right now - basically a straight stick that is about 6-8' tall (see picture - this is not our yard). So based on that, the trees will probably be about the same size for awhile because they have similar froth rates, with the oaks/maple growing faster and will be taller over time.

My hesitation with putting the oaks/maple in the corners is that it will provide zero privacy for awhile. The reason for the 3 conifer groupings was to sort of "break up" the wide open space with pockets of privacy since the conifers will provide immediate privacy. And I was thinking that the oaks/maple would provide some visual interest in between (and eventually some canopy and shade) without needing to provide instant privacy. Attached are a few additional photos of our actual yard.

Any other ideas aside from putting the oaks/maple in the corners? I'm not totally opposed to putting one in each corner (only have 4 oaks/maple and can't buy more) and putting the conifers in a line along the back, but am just trying to think of other ideas that might give us a little but more privacy immediately (since the line will only block out once section of the neighbors and just in that one place)?

Here's a few photos of my back yard so you can see the exact space. The first is the back left corner (when sanding on the patio), ten the middle, then the back right corner and right side.

Middle and right back corner.

Back right corner and the right side (since I'm thinking of putting some of the trees along that side).

The last picture of the right corner and side shows that your neighbors have already planted evergreens. I would take advantage of that and plant your trees on that side and corner of your lot. Then put your evergreens on the left corner. Perhaps this will be more what Yardvaark is advising. His/her layout actually has 4 trees on the right side of the yard.
I've included a link to an article on choices of plants for a privacy hedge. One thought not mentioned is to consider not using the same plant for the entire screen--if something happens to one of the plants, it is almost impossible to fill the gap and make it uniform again.This would be a good place for your idea of planting in groups of threes. Also, make sure you have enough room in that area along the west side of your patio for the mature size of whatever you plant there. There are several different types of arborvitae listed in the article.

Here is a link that might be useful: Privacy hedge ideas