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I would like to say avoid Sony but the A6000 can be had for 200 dollars making it good for you. But if you don't care for video which is one of the Sony's strongpoints I would either bump up the budget to 250-300 or concentrate twice as hard to find a good deal on a used first gen Fuji X-series.
The Sony can be a little bit toy like unlike Fujis and I'm sure you would enjoy handling the latter more.
Find a used body for 200-300 dollars, slap a 30-50 dollar vintage lens on it with an adapter and you are set. If and when you get more into photography you can buy more and expensive lenses.
Remember that you can always sell used things for the same amount you paid and camera gear is best purchased used as so many people buy expensive things new and never use them making them affordable to you.
Fuji X-E1, X-T10, X-M1, X-Pro1, search for everything with X and 1.
