I'm not sure you know what you want. You say you want an interchangeable-lens camera, but your budget is very limited, you want it to be compact, and you want excellent picture quality. Do you really want an interchangeable-lens camera (I'll call them DSLRs, even though it now includes mirrorless bodies)? Or do you just want a fixed-lens point-and-shoot which is cheap, compact, with excellent picture quality?
The usual way people "get into" photography is to go from the cell phone camera to a point-and-shoot which gives you most of the additional control and somewhat higher image quality you're looking for. The step up to a DSLR is a huge one - much larger camera, much more flexibility, much higher image quality, much more complexity, and much higher price. Are you sure you want to skip this middle step?
While it's possible to buy a single super-zoom, stick it on DSLR body, and shoot with it all the time that way, it's a big waste of money. The whole point of buying a DSLR is so that you can change the lenses. The super-zooms, especially to fit a $500 budget for camera + body, are very low quality. You see, the better picture quality of DSLR comes from the significantly larger image sensor. But the larger image sensor requires a proportionately bigger lens to yield the same image field of view. And the price of a lens scales with its surface area (how much grinding is needed to fabricate the lens). So DSLR lenses get expensive really quickly.
If you seriously get into photography with a DSLR, you're probably going to end up with a wide-angle zoom, a wide-aperture fixed focus, and a telephoto zoom as a minimum. A good set of those lenses alone will run you over $2000. Cheaper versions of the set will cost you around $500, but the image quality will be worse than most fixed-focus point-and-shoots (except for the fixed focus lens low-light photography). If you seriously get into it, it's easy to build up a collection of lenses costing $5000 or more. And that's not even including the body yet. A cheap body will be around $300, with the high-end bodies topping $3000.
As mentioned, because the lens mounts are proprietary, if you decide to get a DSLR, you have to pick your system now. Canon, Nikon, Four-thirds, or Sony E-mount (the A6000 is an E-mount). Once you start investing in lenses for that mount, you're committed to that system. Switching systems means you have to sell all your old lenses and get all-new ones, not just switch the body.
So are you sure you want to get a DSLR right now, while you're still learning? This used to be a lot easier in the film photography days. Canon and Nikon were the only choices, and the rate of technological progress in bodies was very slow. So you could buy a body with a cheap lens, and spend a decade or more collecting lenses for it (using the same body for a decade or more). But today new bodies gain new features much more quickly. If you're still in the "learning what a camera can do" phase, any body you buy today will likely be outdated and worth very little in 3-5 years. So getting a DSLR when you aren't yet sure what you want will be a lot like getting a fixed-lens point-and-shoot.
Unless you're absolutely certain you want a DSLR now and are ready to pick a lens system, I'd suggest getting a cheap point-and-shoot as an interim step. Something like a Panasonic Lumix DMC-FZ200 or FZ300 or used Canon Powershot G. That has most of the controls you'll find on a DSLR, plus RAW capability to introduce you with the photo processing workflow (shoot raw instead of JPEG, organize and do some simple processing of your photos, create JPEGs). It'll help you learn what all the controls are for, what lens focal lengths you like to shoot at, what sorts of features you like to use or don't care about, which will all help you make a better more informed choice when you're ready to upgrade to a DSLR. At the rate new features are coming out on bodies, by the time you outgrow the interim camera, the money you lose will be about the same as the depreciation on a DSLR body. So it won't cost you any more than getting a DSLR now (unless by some miracle you happened to pick the exact DSLR and lens system you wanted for future use).
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