One thing that always stood out to me about fresh/real chicken eggs I got from my father was the color of the yolk. Much, much darker than store bought, though I don't know what even plays a role determining that.

Hoping to get a coop up later this year and start experimenting!

Yolk color is a function of xanthophyll in the diet. Backyard chickens tend to get food scraps that contain more xanthophyll than there is in commercial feed, which is why backyard eggs yolks tend to be more orange. If you want more orange in your yolks, you can feed your chickens stuff like marigolds or cooked carrots.

We'd feed birds crab shells every second day for a week or two. When the hard boiled eggs were cut open we'd have concentric red/orange rings in the yolk.

Or you toss in beet peel scraps.

The color of the yolk depends on what the chickens eat; it doesn't really indicate anything about the nutritional content of the eggs. Supermarket "organic" eggs that have a dark yellow yolk might just mean that their chickens' diet included pigments that caused that. A true free-range chicken that had a more wild diet including insects and seeds other than wheat or corn will also tend to have eggs with a darker yolk.

>...though I don't know what even plays a role determining that.

Age is definitely one big factor. The longer they sit the lighter they get. But I suspect diet also plays a big role, and to a lesser extent, lifestyle/overall health

Oddly enough you don't want your eggs too fresh. I remember my grandmother saying that you can to keep them for a couple of days before eating - something about the texture of the white

I think that's only for making hardboiled eggs.

The whites don't separate from the shell well if they're too fresh

According to that criterion the eggs I buy are apparently fresh, though I usually wish the whites were more runny when I use them.

No one likes a non runny white.