A couple of years ago I decided to upgrade from my point and shoot camera in order to take better pictures while hiking.
I think you should ask yourself some questions before you make a decision what camera to buy:
1) What kind of pictures do you want to take? Mountain views? Wildlife, including birds? Pictures of yourself while hiking solo? Night sky?
2) How much do you care about weight & size of your camera? How many pounds is too many? How will you carry your camera?
3) How much do extra controls on the camera matter to you? Do you think you will be motivated to use more advanced modes to take better pictures?
4) Will you be fine-tuning your pictures with advanced software like Adobe Lightroom or similar?
5) How much are you willing to spend on your camera?
Here are some of my thoughts on these questions:
1) I think almost any camera on the market today should be able to take pretty good pictures of the mountains given good lighting conditions, and you could probably stick to your iPhone for this. You may want to pay attention to camera sensor aspect ratio - some cameras have senors with 4:3 aspect ratio, so if want to take "panoramic" pictures with 3:2 or 16:9 aspect ration you will have to give up some megapixels.
The situation gets a lot more interesting when it comes to photographing wildlife. Wildlife often moves fast, so you should have quick access to the camera, fast start-up time and low shutter lag. Some cameras offer continuous focus tracking which will allow you to lock focus on a moving animal or bird. Additionally some cameras allow photographer to manually adjust focus which comes very hand if the animal hides behind leaves or branches. I also feel that having a viewfinder helps me a lot to take better wildlife pictures. Finally, bigger zoom can come in handy if animal or bird is far away.
If you want to take pictures of yourself while hiking solo then the simplest thing is a timer but it requires you to run and take position for every picture. It is much more convenient to use IR or bluetooth remote if your camera supports it. You probably want to have a lightweight mini-tripod for these occasions.
For night sky pictures you will want to have longer exposure times and control over ISO settings + remote shutter control (or use timer), so that your pictures do not come out blurry.
2) My current camera weighs nearly 1 lb 12 oz with the lens, and after a few backpacking trips, I am already thinking hard, how to slim that down. Also if the camera is very bulky and you stow it away in your pack then most likely you won't take to many pictures and you will miss on many wildlife encounters.
3) While I am an amateur by all means, I find that over time I got more comfortable taking control of my camera and for example I tend to verify / fine-tune focus more often now then before I got my camera. I also like to defocus some pictures by lowering F-number in Aperture mode.
4) I use Adobe Lightroom to fine-tune some of my pics, however, to take advantage of full capabilities of such software you need to shoot in raw format.
5) You already said that $500 is the limit which eliminates most DSLRs and many interchangeable-lens cameras.
So what is my current setup? I have Sony a6000 with 18-200 mm lens which I consider sufficient for my needs on the trail, and for the record, I don't want to carry any more lenses with me. I also have Pedco UltraPod II mini-tripod and Photo & Tech IR Wireless Remote. I am pretty happy with the pictures and don't have much to complain about besides weight (although I could point out a few things if anyone cares to know.)
What would I like to have: driven by weight I would like to upgrade to Sony DSC-HX90V:
https://www.sony.com/electronics/cyber-shot-compact-cameras/dsc-hx90v Why? Sensor aspect ratio is 3:2. Small size. Zeiss lens with large zoom. Focus can be adjusted manually with a ring at the base of the lens. The camera has a pop-up Electronic View Finder. I generally like pictures I get from my Sony cameras. It appears that this camera can be controlled with a remote (RMT-VP1K.) What's not to like? It does not have an option to save images in raw format.
Given that my camera budget right now is pretty limited, I am far more likely to upgrade my smartphone to Sony Xperia XZ that has a 23 MP camera with built in optical image stabilization.
I suggest that you read some user reviews for whatever camera you pick before buying. Good luck!