At this price you should be able to move into a camera with a substantially larger sensor and very good quality lense (oe lenses). However, keep in mind that many of these cameras use interchangeable lenses. You will spend more if you become interested in adding a couple of lenses.
Canon offers the T5i and the T6 in this price range. You might like the Canon SL1, a smaller and lighter SLR body that uses the same series of lenses that the T5i or T6 cameras use. All three of these cameras are excellent. I would recommend the SL1 or the T5i. These cameras have a large sensor and Canon lenses are very good.
Nikon offers the D3300 and D5300 models in this price range. The D3300 is designed to be easy to use and is ideal for a first-time SLR camera user. The D5300 adds the ability to send images across WIFI to a computer or tablet. Both are excellent cameras with very good line-up of lenses to choose from.
Nikon also offers a high quality point and shoot camera. It is smaller and has a sensor that is bigger than most cameras of this type and provides a built-in 24-85mm lense that lets a lot of light in. This is the Nikon DL24-85 model and it will allow you to take photos in a wide range of lighting conditions.
Fujifilm offers the X-A2 model. While this is a decent camera with a three to one zoom lense and a large sensor and the capability to take excellent photos, I do not recommend it very strongly. That is because it has no viewfinder and the lense is fairly slow (it will work great in good light - not quite as well in low light). If this won't bother you it takes very nice photos in good light.
Sony offers the a6000 camera. This is an excellent camera (so long as you do not intend to buy several additional lenses for it). It comes with a three to one zoom lense and there is one additional affordable quality lense (a 55-210mm zoom). Which is probably all you would really ever need. Other lenses are very expensive for this model. Now, that said - I highly recommend this camera. I believe it to be the best model in this price range (Nikon would agrue, but I like the Sony a6000 over any Nikon under $600.00).
The Panasonic ZS100 (in silver) is just under $600.00. This is a very good camera. It has a medium sized sensor and a ten to one zoom lense. Photo quality is excellent. This is a point and shoot type camera (no changing lenses). The size is nice - not quite pocket size, but still very small and light.
The Panasonic GM-5 is a small camera with the ability to change lenses. It uses a four thirds sensor (which is bigger than medium and provides excellent quality images). This model has a viewfinder and is easy to carry and use. Good all around camera.
The DMC-G6 is a similar camera (to the GM-5) but is larger for someone who wants a camera with some size to it (can be easier to hold for some people). This is a almost great camera that uses the four thirds sensor.