I currently have a point and shoot nikon p510 with 42x zoom. I actually use the zoom a majority of the time for wildlife photography (mostly birds). The only thing holding me back currently is image quality. Sometimes when I go to crop a photo the pixels are noticeable. I was looking into getting a nikon d5200 but I am really open to any camera around a similar price range. Also if I want to get the same zoom capabilities of my point and shoot, what mm lens would I need, and generally what is the cost of that? Thanks in advance!
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.
Just me, but for birding, if that includes bird-in-flight photography, take a look at Canon as well as Nikon. Particularly if you can only afford an entry-level body. Because Nikon's D3xxx and D5xxx bodies will only autofocus with an AF-S lens. And you'll probably want at least 400mm if you're shooting birds. Now price out the Nikkor AF-S 80-400 VR, and then the Canon EF 100-400L IS USM (MK I and II) and the EF 400 f/5.6L USM. Nikkor's equivalent to the MkI 100-400L is AF, not AF-S, and therefore doesn't autofocus on a D5xxx.
400mm generally means a minimum of $1000+ in glass. But on the Nikon side of the fence it's more like $2500 minimum, so most Nikon wildlife shooters on a budget will end up with a Sigma lens. On the Canon side of the fence, you got a few more choices.
And it took me about two years with an entry-level Canon and a 400/5.6L to get to the point where I liked my bird photos. :)
It will be impossible to get this kind of zoom range using a single lens on an SLR. On a APS-C SLR like the D5200 you will be able to cover the range with 2 lenses. There are a number of options for zoom lenses in the 1x-2xx mm range ($200-$400)and there are at least 2 options available for 150-600mm which would cover you for the wildlife photography. The 150-600mm lenses should be around $1100. So you would be looking at about $1500 for the lenses. Quite a jump in price from the P510 but also a jump in performance.
Also keep in mind that a 150-600mm is a large and heavy piece of glass, about 2kg. To shoot birds in flight a tripod (with a gimble mount) is highly recommended.