Labrador_Remy: You are non-specific of your budget, but for just overall picture taking, without too much muss & fuss, I would go for a camera in the Sony RX100 line. I have a RX100-7($1,300), and it was expensive, but for me worth every cent. The Sony RX100-3($750) is less expensive models that may satisfy your needs and is still in production. But the drawback is that the Sony menu system is kind of complicated, and takes a bit of work to figure out.
I think the similar Canons have competing models, but they don't have the durability, I had a Canon S100, and it only lasted like 3 years, Canon repair it under warranty, and it later developed the same problem, again. So far my Sony has lasted way over what the S100. BTW when you have to personally pay for camera repair it is better to buy a new one.
Don't know about the Fujifilm cameras, since I have never owned owned, but from what I hear on the grapevine, it is really good optically, don't know about the mechanical issues.
ETA: The phone cams have their place, but the tiny sensor really shows it shortcomings, even though Apple spent enormous amount of develop money on it. I believe the Sony 1-inch sensor has 4X the area of the Iphone sensor, so your photos and videos will improve dramatically especially under less than idea lighting conditions, but you have to remember to bring your camera, whereas you already have your phone with you.