>>2643260a6000 is $600+ with a kit lens. a5000 is more in your budget if you're going with Sony.
I recently bought an a5100 and I like it. It is pretty compact, but it's not tiny. Battery life is good and supposedly lasts ~400 shots. I charge it before I take it with me and haven't gone under 50% yet. No idea about lenses, since I just bought it to learn photography. Picture and video quality is good.
It's got way more bells and whistles than I really need, but weather proofing would have made it perfect.
If you're looking for an entry level camera, consider a Pentax K50. Weather proof and there's a kit that comes with a couple of WR lenses all under $500. That's what I have my eye on next. Just took my camera on a kayak and was very afraid a tiny drop of water would ruin it.
