a Full frame DSLR is very different to an Iphone camera.
Firstly weight...it is much bigger and less convenient. You will not want to carry it everywhere. The entry level DX bodies like D3200 and D5200 are much smaller and lighter but full frame has a few advantages explained below.
Secondly light sensitivity - given your stated interests - there are quite a few that involve low light photography. If you are not going to invest in the top pro lenses which are big and very pricey but let more light in, then full-frame is the next best thing because full-frame sensors have way better low light capability then DX/APSC, Nikon 1 series or the other small micro-four thirds cameras like Panasonic G series or Olympus OM-D (as good as they are)
Obviously you will get better image quality with the DSLR compared to iPhone but it will also have significantly different depth of field and the photogrpahoic style will be very different.
As a DSLR newbie it would be advisable to get a camera with Scene modes so D600 or D7000 (DX) are both good options and both have sufficient manual controls to permit creative options as your skill increases. Nikon seems to have invested more time and energy in its full-frame line lately.....a full range of entry, mid and upper full-frame zooms is now available, along with a lot of primes (fixed focal length lenses).
So i'm confirming your initial choice of full-frame. If you had said wildlife and birds it would be a different story.
Now for lenses : I'd advise to start with an entry level or mid range zoom. The entry level is the 24-85mm VR and it is a cut above the DX entry level lenses. It is available at lower cost in a kit with the D600. However, given your budget I would suggest the 24-120 f4 Zoom. This is the midrange option. It offers more light and range and will cover just about 80 to 90% of pics that most people will take. It is a bit bigger and heavier than the 24-85. Image quality is excellent.
After this if you want a wider zoom, the new 18-35mm just announced should be a good option at reasonable cost. For a longer zoom the 70-300 VR is probably the best value for money Nikon zoom ever. However, I suspect that for street photography, especially in poor light, you will want a smaller, brighter and less obtrusive lens and here the 35mm f2.0D-AF or 50mm f1.8G lenses offer excellent image quality at low cost and they let in up to 4 times more light than the entry level zooms.
D600 offers all the options and controls required for starting with automatic scene modes and growing progressively into complex creative use of manually selected settings. Enjoy!