Hi Luke.
I upgraded from the 3100 to the 5300 two years ago. I am very happy with the 5300. I use it piggy back with the 18-55 kit lens, and also through my telescope with quite good results (you can find my pics in the photography section)
The only difference I can see between d5500 and d5300 is the screen is touch on the d5500 (same sensor, processor etc)
http://www.gizmag.com/nikon-d5500-vs...compare/35653/
So if you want to save some money maybe you can find a second hand version of the d5300.