Do you have concerns over size and weight? The main thing that many people new to DSLRs want is something relatively small and light - for example, the D7xxx series is quite big and heavy (personally I prefer that as it makes for a more stable shot and is more comfortable for me to hold but many people prefer small/light).
The D3300 I think has a later/better sensor than the D5200. If you don't need the articulated screen then I would go with the D3300 out of the two.
Get a good kit lens like the 18-105mm or the 18-140mm and you can do a lot with it. The 16-85 is slightly better quality but doesn't have quite the same reach. Lots of people like the fast 35mm or 50mm primes but if you are just starting out I would avoid primes until you are more used to the camera (I have the 35mm and never use it).
Lens choice will depend on what photography you are doing. I went through all sorts of lenses and mostly these days just use a Sigma 17-50 F2.8 with occasional swap to the Nikon 70-300 when I want much more zoom.
I have to say I like the Nikon cameras but haven't used any other make.
One thing, if you are hanging the camera around your neck then I wouldn't use the strap that comes with the camera. I bought a strap that uses neoprene which makes it so much more comfortable to carry the camera. Many people swear by the black rapid straps but I haven't used them so can't comment.
Finally, the on board flash is useless with many zooms - can cast a shadow on the image. If you are going to need flash you might want to get a proper flash. There are some 3rd party flashes around that I have seen many people use on forums, the Nikon ones can be a bit pricey for anything decent.
If you want to get lots of thoughts then either dpreview or nikonians.org are good resources for discussions around Nikon cameras (and there are many threads asking for opinions between different models).