Jimmi I'd recommend going to a shop and getting a feel for different DSLRs first, as compared to what you're used to, it'll feel really bulky. I've got the D5100, which will be broadly similar to the D5200, and although bulky, it feels comfortable. 

I can take fast shutter shots during the day, and I never have issues with blurring. Even if you don't want to piss around with settings, you have the benefit of sport mode, which will allow you to shoot buses in motion. I know with my D5100, the shutter speed can be adjusted to perform a multi-shot rapid fire, so you can just compensate by taking 5-6 pictures for each shot.

You obviously don't get an optical zoom with a DSLR, but I take all my photos with the same 18-55 lens you'd get with the D5100. Dan will have the same lens with his D5100 too. Just take large photos and crop them down, and you'll have no issues. 

Both the D3200 and D5200 are great prices, and I'd be happy with both, but I'd push towards the D5200 if you can afford to. The lens is much more capable when it comes to autofocus (39 point as oppose to 9), and the ISO sensitivity is better.