The Nikon P510 is what usually is referred to as a bridge camera. That is a camera somewhere in between a point-and-shoot and a DSLR. It has got a zoom lens, but the term zoom does not mean that you will get a narrow angle of view (close to the subject). It means that you can adjust the focal length of the lens and therefore the angle of view (how close or far away from the subject you'll get).
The lens that it uses corresponds to a 24-1000 mm-lens on a full frame camera. In order to cover that range with a DSLR you would have to use multiple lenses and the longer ones would be very, very expensive.
Since you're looking for a DSLR with aN APS-C lens you will have to take the crop factor into account when determining what lens(es) you'll need. 24 mm corresponds to a 18 mm-lens on an APS-C camera and to get a focal length of 1000 mm you'll need a bit over 600 mm and that lens costs around $10 000.
Of course you can use a shorter lens (a lens with a shorter focal length), but then you'll have to crop the photo.
Even though the lenses the lenses giving you the same capabilities as the P510 on the long end are very expensive, a DSLR might still be useful with a shorter lens (there are 18-300 mm lenses, corresponding to 24-450 mm due to the crop factor that cost around $700 and given that a cropped 300 mm APS-C image to a corresponding 1000 mm on a P510 will look a lot better it might be worth it. The DSLR:s also has a lot better performance in autofocus and I suggest that you'll try one out in a store with a longer lens to learn the differnce.