I'm trying to interpret some of the responses here because I know next to nothing about photography, but
No matter what camera body you will use , key is the lens and knowledge how to use the camera.
A phrase often used when analyzing the problem behind a poor picture is that "the problem is behind the camera".
With what you seem to have access to right now you can learn photography, no matter what your subject it.
No shortage of video tutorials on youtube how to use dSLRs in ways other than setting everything to automatic, and how to get the camera to autofocus on a subject and stay there.
For many starting out seeing the pictures they can be a brutal reality check that leads to frustration. Movies and TV shows (usally thrillers) give the wrong impressions what long lenses can do, e.g., taking detailed photos from a distance. The photos of birds published here and elsewhere are examples where a lot of know how and experience add as much to the result as the gear that was used.
In the real world one still needs to get pretty close to a bird to get enough of it to fill the frame even in part. With the best 500mm prime lens money can buy, to get a small bird into a frame for a decent looking image and without having to crop like a maniac one needs to get pretty close to the bird (~10m for a typicla size songbird). Birders are more used to think in magnification and as other folks already pointed out even with a long 500 or 600 mm lens we get a view that is similar to what we see through 10x binoculars (before our brain starts zooming in).
There are some 150-600mm lenses from several different manufacturers that will provide reach without having to break the bank, even more if they are 2nd hand. Putting money into a decent lens to use on an older body gets you further than getting a new body and use it with an existing old short lens. Its the lens that gives you the "zoom in", not the camera body.