Currently, I hate running.

slow the fuck down.

My lungs hurt.

slow the fuck down. Walk if you have too till you get into shape.

I have to breathe out of my mouth

Most runners mouth breath. Get some gum for the dry mouth.

and I guess I just don’t have a “runner’s body.”

you are human, you have evolved to run long distances better than any other animal on this planet.

but I have no idea how to build healthy running habits

run every day. Run slow. Listen to your body most of all. Your body will tell you what is wrong and how to fix it if you listen. (I know it sounds like hippie bullshit, but it's basic troubleshooting and problem solving skills).

push myself to be better.

You need to run slow. Seriously. Run slow.

You need to build endurance and fitness and the only way to do that is by putting on a lot of "easy miles". Once you are putting on 30 to 40 miles a week you can work on speed or strength or drills; but at the start, just focus on building your core foundation with logs of slow easy miles.