This is going to be a long one! Get ready!
If you could reply back after reading this post with more detailed info about your needs.
What are your needs for:
-Portability Size, lightness, etc. Do you need a camera that can go everywhere, or will it primarily be for special occasions? This is a compromise section. A larger camera generally will have better quality for less money than a compact camera with excellent quality. There are cameras however that can satisfy both.
-Focal Lengths. Do you need multiple focal lengths (zoom levels), or are you okay with one? Generally, single focal length lenses will be extremely sharp in comparison to zoom lenses. A 50mm lens will be significantly sharper than a 18-55mm zoom lens set to 50mm for example. That's not to say you can't take great photos with a zoom lens.
I've got a few more recommendations that I can give based on your next reply. These would include compact cameras, a couple mirrorless cameras. My qualification to give you recommendations is that I use to be heavily into the hobby, and even worked on some professional jobs at one point. I really enjoy cameras nonetheless, but I am not as serious about it as I once was.
Okay. First of all. I would consider buying second hand, it's absolutely great. Find a great condition model and buy it with the depreciation already hit. Spend the rest on a good lens.
So. My two recommendations, I have researched for and decided to purchase. I don't have either anymore, simply because I bought a more expensive camera. They were great while I owned them. One is a DSLR, so nothing new there. The other is a mirrorless.
I used to have a Nikon D5200. You can find out under $500 brand new. Match that with a 50mm f1.8 Nikon lens for about $150 more, and you will have a nice setup. You will get very sharp photos, but will be stuck with one focal length (no zoom). You can also go for a zoom lens if you want multiple focal lengths. A quick look on eBay sees these cameras going WITH lens for ~$350, in great condition with lots of photos for you to look at. Honestly, this camera gives great quality, photo and video.
Another, cheaper option is the Sony (sorry) NEX-5N is a great option. A quick search on eBay shows good condition ones going for ~$250 with a lens. The image quality isn't quite as good as the Nikon, but it certainly is no slouch either. It is very small, very light, and is super easy to use. It doesn't use those stupid Sony Memory Sticks, it runs a basic SD card. The video quality is good, as is the picture quality. Not as sharp as the Nikon though.
There is also the Sony A5000, which is an updated version of the NEX-5N with a LOT of improvements. A bit more expensive however.
Some other models for you to research:
Canon Rebel T5