It's All About the Lenses
When you decide to jump in and buy a better camera, the easiest thing in the world is to obsess over the camera body you're buying and spend your entire budget on it. Bad idea.
The thing that makes DSLR or mirrorless cameras amazing is the multitude of lenses you can use with them. The lens that comes with your camera will likely be mediocre at best; it's the prime and zoom lenses that make your camera a system you're buying into. That's the best reason to choose a specific camera from Canon or Fuji or any other company: there's a whole collection of lenses specifically for those cameras, and it's the ability to use those lenses that should drive your camera purchase.
How much of a difference do lenses make? Go check this article from Josh Ginter, which does quite the good job showing the differences in pictures just from using better lenses.
Surprising, right? And yet, as Ben Brooks reminds in his reply piece, great lenses don't always have to cost a fortune. That's actually one of the main reasons I bought a DSLR instead of a mirrorless camera: there's tons of lenses for Canon DSLR cameras, with many affordable-yet-great options. And, with so many DSLRs out there, there's a wealth of quite good used lenses, like my 10-22mm wide lens that I bought used for less than a third of its standard retail price.
You have to buy a camera to start with (that is, if you're sure you really want to jump into interchangeable lens cameras), but lenses should actually be the first thing you're thinking about. That's where the great pictures come from.
Well, actually even that's not true. As Ben Brooks said, "The most important part of any camera is you." But you'll be looking through the lens, so make sure it's a good one.
Now just try to not get addicted to buying ever nicer lenses…
Thoughts? @reply me on Twitter.