You're probably (or maybe not, see edit below) looking for a "mirrorless" camera, which is like a DSLR with a digital viewfinder instead of the traditional optical viewfinder. (IMO, this is an advantage and there's no reason to go DSLR anymore.) Like a DSLR, the mirrorless camera lenses are also interchangeable, so you can grow the abilities of your camera as you learn more. The low-end options you'll find for around $600 new will typically come with a short zoom lens, and the overall package will be a nice step up from a smartphone, even if you're shooting in full-auto mode, but you'll gain more if you take a little time to learn the partially-manual modes, like shutter/aperture priority.
There are a handful of mirrorless systems (they differ per camera manufacturer, and some manufacturers make more than one system), and which system you choose will probably take some research/asking in a specialized community. It really does matter which you choose, because lenses are generally not swappable between camera systems, and lens pricing is all over the place. If you know photo enthusiasts in real life, it's not a bad idea to go with the same one they use, as you might be able to borrow lenses, accessories and knowledge from them.
Not sure how Vietnam works, but buying used is often a good choice and will get you quite a bit more for your budget if you can buy from a trustworthy marketplace. Starting off with a camera that's a few generations behind probably isn't a bad choice at all, as the tech evolves fairly slowly and will save you some money if you realize down the road that you bought into a camera system that doesn't best suit your needs.
edit: It might also help to find out what kind of nature photography you envision yourself doing. I read "nature" and thought of wildlife (and assumed that chances are you'll eventually want a longer zoom), but for all I know you might be thinking something like landscapes, which is typically at the other end of the lens spectrum. If you're sure that you've outgrown your phone, there is another type of camera, basically an advanced point-and-shoot, which feature large-ish image sensors and offer quite a bit of manual control, but no interchangeable lens (just a decent zoom). They're impressive for how much camera power can fit in your pocket, and you'll find options around your price range, but these offer very limited, if any, options to grow in the future.
Agreed. Pants that have sat on public benches, chairs, etc. are okay to go sit on more public benches and chairs another day, but them hanging out with clean clothes is not the same thing.
You're probably (or maybe not, see edit below) looking for a "mirrorless" camera, which is like a DSLR with a digital viewfinder instead of the traditional optical viewfinder. (IMO, this is an advantage and there's no reason to go DSLR anymore.) Like a DSLR, the mirrorless camera lenses are also interchangeable, so you can grow the abilities of your camera as you learn more. The low-end options you'll find for around $600 new will typically come with a short zoom lens, and the overall package will be a nice step up from a smartphone, even if you're shooting in full-auto mode, but you'll gain more if you take a little time to learn the partially-manual modes, like shutter/aperture priority.
There are a handful of mirrorless systems (they differ per camera manufacturer, and some manufacturers make more than one system), and which system you choose will probably take some research/asking in a specialized community. It really does matter which you choose, because lenses are generally not swappable between camera systems, and lens pricing is all over the place. If you know photo enthusiasts in real life, it's not a bad idea to go with the same one they use, as you might be able to borrow lenses, accessories and knowledge from them.
Not sure how Vietnam works, but buying used is often a good choice and will get you quite a bit more for your budget if you can buy from a trustworthy marketplace. Starting off with a camera that's a few generations behind probably isn't a bad choice at all, as the tech evolves fairly slowly and will save you some money if you realize down the road that you bought into a camera system that doesn't best suit your needs.
edit: It might also help to find out what kind of nature photography you envision yourself doing. I read "nature" and thought of wildlife (and assumed that chances are you'll eventually want a longer zoom), but for all I know you might be thinking something like landscapes, which is typically at the other end of the lens spectrum. If you're sure that you've outgrown your phone, there is another type of camera, basically an advanced point-and-shoot, which feature large-ish image sensors and offer quite a bit of manual control, but no interchangeable lens (just a decent zoom). They're impressive for how much camera power can fit in your pocket, and you'll find options around your price range, but these offer very limited, if any, options to grow in the future.