I've gone totally Sony mirrorless, from Nikon DSLR. For years I've had a Sony a6000 (interchangeable lens mirrorless) which is small but mighty in terms of features. There are more recent/advanced/expensive models out now, I think the a6500 is the latest and greatest. But a lot more expensive. The a6500 has among other things in-camera lens stabilization and a fancier sensor. I now also have the bigger/pricier (!!) full frame Sony A7R2 and I love it but I still take my a6000 out shooting especially when I want a lightweight rig to carry around town.
In the price range of $600, you can get the a6000, new, with the 16-50 kit lens, or just the a6000 body alone for about $550. I've seen it bundled with the 16-50 mm and 55-210 mm lenses for about $700, new. I think there may be good deals on used ones (i'd stick with reputable dealers like KEH, B&H Photo, Adorama). EDITED to add: buy at least one extra battery and a plug-in charger, don't need Sony brand, something like Vivitar will be fine. The Sony mirrorless cameras are famous for short battery life, I'm used to it and it's just something I deal with.
What I really like, it that you can use adapters to mount your older lenses - regardless of make - to the a6000. I still own a bunch of old Nikon manual focus lenses. I use inexpensive adapters and mount them on my Sony mirrorless cameras. The camera will meter the exposure just fine in M or A mode, and Sony has a "focus peaking" option that uses color in the LED display so you can fine tune what your lens is focused on. Makes manual focusing, IMO, a breeze.
An advantage of the a6000, IMO, is that the sensor is a lot larger than regular point and shoots (although the Sony RX-100 cameras have a 1" sensor which is better than regular P&S). If you're used to the images from a DSLR, depending on how you tend to edit and what you do with them, you may be disappointed with some point and shoot cameras. Here's a rundown of digital camera sensor sizes.
There are no doubt good deals on other makes of mirrorless cameras, but I'm only familiar with Sony.