-
If you're not set on a DSLR style camera and open to other brands, then I'd take a look at the Fuji X-E2 with the 18-55mm lens.
Although I'd agree a prime lens is better every time. The X-E2 with the 27mm 2.8 makes a fantastic camera. Really light, great for travelling, and the image quality is on point.
I'd always recommend mirrorless over DSLR for someone who wants to move from phone photos to a camera with high image quality. Worth a look for sure.
*replied to the wrong message soz
The RX100s have a 1" sensor, whereas a DSLR will have a slightly larger aps-c sensor, which means a shallower depth-of-field can be achieved and, all things being equal, should be better in low light.
The autofocus will pretty much always be better on a DSLR as well...
A D5300 will be great. The kit lens is fine but I'd recommend a fast prime to compliment it. The 35mm 1.8 is cheap and very good. It's a good general purpose focal length, especially for people and will get some nice, shallow dof.