When I was a child we had a sun filled yard. We planted a number of trees for fun and over the years the trees have grown and crowded each other out. I'm looking for some advice on which one(s) to remove to make the yard more balanced and to grow some grass (currently we can't grow any grass and the soil has begun to erode). The pine tree in the middle (left) seems to be the heartiest, but also is blocking the most light because it is on the south side of the yard.
I'd like to know what the broadleaf, deciduous tree on the right actually is, can't tell from the pic.
If you wanted to create a proper garden and have some good grass, and you were in the UK and I'd been called to do a redesign, from what I can see just in the photograph, I'm afraid I'd unhesitatingly propose removing all the trees, with the possible exception of the one in the foreground on the right (depending what it is). I'd replace one, two or three trees with something a bit more decorative and generally smaller than what's there, and add other planting. But then you're not in the UK and you're not asking for a redesign. Even so, removing one or two on the left (south side) would seem to be appropriate in order to grow any grass at all, and lessen the general 'untended forest' appearance of the area. The trees on the right will have been deprived of good light levels simply by the proximity and height of the evergreen trees on the left - if you took out the ones on the left, you can expect more vigorous growth on the remaining trees. Which might or might not be a good thing, again, depending what they are.
They're so cute when they are small...and then they get big.
Personally, I'd cut any pine/evergreen if you are looking for light and grass. While you could have it limbed into a pom-pom/lollypop to let more light in, it's not a good look for them, and they are otherwise more-incompatible with those goals than other types of trees.
If you didn't specifically want grass, but just to have green cover and stop eroding, myrtle or pachysandra can do well under pines (and other trees) but don't take traffic too well.
Reducing the other trees to more of a single-stem form (as opposed to the "grown in the open" bush/multistem form epitomized by the leftmost tree) and removing lower branches would be a large start on letting more light in, but may also be expensive to have done in such tight quarters (looking at "narrow strip" nature of the yard and neighboring fences.)
Keep the trees, you have everything you need for a green lawn right there!
Looks like you have a healthy dose of beautiful moss already growing. All that's needed to "moss up" the other area is a little raking, and some transplanting to hasten the process.
The moss will spread to bare soil, but it's not invasive. For transplanting, take a patch from the established moss, about 2"x2". In the new area, tamp down the soil, then lightly rough up the surface. Next, lightly brush away clumps of soil from the bottom of the moss patch. Mist both areas, then press the moss very firmly into place (you can step on it gently, but don't mash it in too hard).
The moss will spread, transplanting helps, and for even more help you can purchase a 25sqft grab bag of moss online fairly cheap (a lot cheaper than having a tree removed) and/or gather moss.
Gathering: moss is part of the ecosystem, so be gentle. Taking a 2"x2" patch from a 1sqft patch is fine, and in a deadwood-heavy forest you can take a small portion of moss-covered bark from deadwood or even some moss-covered fallen branches. Filling a 1-gallon ziplock on a short walk is easy.
Best part is what you can do b/c you don't have to mow the area. I mean, not mowing is okay (I actually don't mind mowing), but because there's no danger from the mower, you can put shade-loving plants/ferns wherever you like.
There's also a variety of shade-loving groundcovers you could install. I'm a moss hound, so that's the first place I go, but those others would work as well. I had a batch of "shade-tolerant" grass fail on me once, and that - coupled with many folks telling me that there are no "shade-loving" grasses - got me to moss, which has developed into a full-fledged obsession. I'm not proposing you get addicted, but it IS an option and it's a wonderful one, imo.
I'd get rid of the front left tree in the picture. Might not give you more light than getting rid of one of the others, but it will make your yard look and feel more open.