I have the D3200, as said, it has a great sensor, and while prices of 2nd hand have come down since the D3300 was released, it's still holding its price well.
I had similar budget to get my Daughter kitted up to finish her GCSE course, and start her A-Level, earlier this year.
I plumped for Second had D3100 for her, as best VFM.
It doesn't have the infra-red remote release, or quite the same pixie count, but the extra pixies on the D3200 are oft let down, I think, by the kit lenses, that cant really exploit them, so small odds, certainly to a beginner.
D3100's still packing more than you need to be getting on with, and bodies cheap, lenses 'spensive.. saving allowed me to partner her camera with the 'better' 35mm f1.8 lens instead of the kit 18-55, and still have a bit of change for a decent memory card...and that glass more than makes up for the difference, I think, and will be even better on anything she might upgrade to later.
Mentioned the IR remote.. neat toy if you have a tripod.. not much use elsewise.. and even though they made a point of selling the 'twin' IR sensors on the camera, one front, one back, so that you can be infront of behind it to fire by remote.. it still isn;t all THAT reliable if you aren't directly infront of behind! Doing some night time light-painting for her school-work, we got some interesting shots not knowing whether we'd actually fired the shutter.... a 100% reliable cable release, if you want/need that feature is only about £7 off e-bay.... so, saving on a D3100 over D3200 could leave enough the piggy for tripod and remote release, AND a 'fast' SD card.
D3200's higher sensor res makes BIG picture files. This begs a bigger capacity card, and means that you are likely to want a faster write speed. The 3100 does write significantly quicker than the D3200, which again was shown on that light-painting shot with the daughter; damn annoying waiting five minutes between shots for to fish 'noise reduction'... especially if you are looking at a black back screen wondering whether the IR remote has actually closed the shutter and it IS doing NR! So another bonus, you can get away with cheaper,smaller/slower SD cards with the D3100..
So, D3100 is more likely to be found in budget, and leave more of budget for useful extras, and for a first DSLR you are unlkely to notice much other difference.
