I can't really tell what it is that I'm looking at in that picture. Orange Peel looks like the texture of an orange and came from the factory with it. It isn't something that just crops up somewhere down the road. If what you are seeing just started appearing then it could be any number of things. Hard water damage, failing clear coat, acid rain, etc. If your clear coat has been etched by calcium or acid rain, then the only way to repair it is to cut the paint. That needs to be done with a high speed polisher, so don't even try it by hand. I suppose it is possible to do it by hand, but it is not possible to do it evenly over the whole panel, so you'll just cause more harm than good doing it that way.
Definitely take it to a professional. Skip the body shops and the dealerships. They know very little if anything about repairing paint. A body shop will know more than a dealership, but they aren't going to polish it out for you, so don't bother with them. Find a reputable detail shop (not just a touchless car wash) and have them take a look. You need to find one that handles cars one at a time and doesn't just put them on a conveyor belt. That is just a car wash. A detail shop is going to look more like a tire shop with multiple bays for working on cars. Car washes won't be able to fix this problem either.
If you are seeing depth to it then it is already etched in the paint. The damage is done and it isn't likely to get worse. However, now you have compromised areas of your clear coat and it needs to be leveled out to prevent a number of problems. Clear coat failure can start with hard water damage, so you definitely want to get this looked at.