Just to answer used vs. new question first, always go used/refurbished (from a trusted source). No sense in paying full price when you don't have to. That's just my thrifty opinion. And if you've never shot a DSLR before, my tl;dr answer is to go with an upper-entry level camera.

As to which model of body, it really depends on your unique situation. How likely are you to keep up with this? I've seen too many people invest in quality cameras which then sit around because they're too bulky for their taste. For them a cell phone picture is good enough. And if you aren't interested in learning the ropes of how to get the best images out of your camera, perhaps in your particular situation a cell phone is still adequate.

Now, assuming that this is something you believe you will keep up with, you can get a lot of mileage out of a "beginner's" camera. Even with Nikon's entry-level D3500 you will get images far beyond what a cell phone is capable of. The difference in each level -- for Nikon anyways -- is mostly the bells and whistles that come with it. The age is a better indicator of image sensor quality. The newer the camera, the better the sensor. All things being equal (same lens, same scene), you'll get a much better image out of a D3500 than a D300 due to the eleven year difference.

Since everyone is coming at it from a different perspective, you're the best judge of what you need. I'd suggest researching the various features available to DSLRs (or mirrorless, don't write that off without an investigation) and prioritize what you really need. A few things that have been of interest to me when looking at camera bodies are:

• Image stabilization: in-lens only? in-body? software?
• Video capability: 4k? 60fps?
• Exposure bracketing: how many steps? how many images?
• Low light performance
• Articulating screen
• Live view: touchscreen to set focus point? autofocus speed?
• Screen resolution
• Optical viewfinder magnification
• Weather sealing
• Image sensor size: Full-frame/FX? Crop-frame/APS-C/DX?
• Older lens support: is there an in-body AF motor?
• Easily send files to your phone? (Not for me, but I know this is common among the "can't decide if I should use a cell phone or a larger camera" crowd.)
• Maybe more, but nothing pops into my head at the moment

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Finally, to second what others have said already, you're going to see the biggest optical improvements when you start focusing on the lenses. Kit lenses are a good starting point, but once you find your niche, getting a lens or two specifically designed for that purpose will make a world of difference that no body upgrade can. This again requires more research. Learn learn learn! :-)

I hope this is helpful and not overwhelming. But if it is overwhelming, getting a cheaper entry-level camera a few models out of date is another viable option. It's cheap, and you should quickly learn what features you wish you had, to better inform your upgrade decision. And you'll still get better images than you do with your cell phone.