What type of pictures are you going to take with your camera? And what form factor would you find acceptable (i.e. how big a camera)? Since you prefer to not have interchangeable lenses and are coming from a point and shoot experience, I'm going to assume that you like using the automatic settings on your camera and don't want to futz with the manual controls. If that's true, you can get large[r] sensor cameras in a P&S form factor which will take nicer pictures over a wider variety of shooting conditions than your typical P&S cameras and iphone. The two I'm most familiar with are the Canon S series (S95, S100, S110, S120, etc) and the Sony RX100 series. The Canons can be had well under your $700 budget, and the older RX100 versions can also be under your budget. Both should give you decent pictures in full automatic mode, and they both give you the option of manual control should decide you want to start playing around with those settings.
I'll also second the recommendation about a photography basics online tutorial. Some basic understanding of photography principles can help with your decision and also improve your shooting technique.