I'd actually advise you to go even lower than that to start with. Go for one of the entry level cameras, or a mid-high end quality mirrorless ILC. And then buy a couple of cheap primes. For Canon the 40mm f/2.8, 50mm f/1.8 are quite cheap and reasonably good quality, especially the 40mm. Not sure what the Nikon decent & cheap prime(s) are.
Over the years I've learned that the easiest way to kill a hobby is to start with crap equipment. Instead of climbing the ladder from cheap bodies and lenses to the good stuff, it makes more sense to get the best suitable stuff one can afford straight away. Especially with lenses there isn't even a huge risk of losing the money if the hobby doesn't last.
That being said, I started my DSLR hobby with a damn Rebel and I bought a better body after 2 months. Should have never bought the Rebel. I don't even like the photos it produced even though it was paired with decent glass. Now I have mediocre quality photos of my first daughter's first months.... Boring DOF and low + grainy image quality already at ISO 800.
And as a final note, my budget basically is enough for either the 70D or D7100, so roughly $1150 or so. Since this is my first camera, is going with either the 70D or D7100 the smart choice, or should I honestly just save up for the Full Frame, like the D600 which is only $1500 currently? My brain hurts guys, help!
How long do you need in order to save the extra cash? Honestly, if you want quality and the lowest possible price, I would look at the Sigma 35mm + a 6D or the comparable Nikon. The types of photography you mentioned all ask for FF. Crop has no benefits there.