Like you, I used a Fuji S5200 before upgrading to a Nikon system. I emphasize the word, "system" since a dSLR purchase is not exactly like a PnS purchase. The camera body itself is just one part of the equation. One should look at lenses and other accessories at least as critically as the camera body itself - perhaps even more critically.
There are many, many dSLR models to choose from, from all the major camera makers. I suggest you read reviews of cameras and lenses that would fall within your budget. If you are wanting to shoot low-light photos without using a flash, you will need to look at lenses with a larger maximum aperture (lower "f" number). In the world of zoom lenses, this can mean a significant investment. Primes (single focal length) are somewhat less expensive. For example most of us Nikon folks started off with a zoom lens or two plus a $110 50mm f/1.8 lens for low-light stuff. The new favorite fast prime, at least among Nikon folks seems to be the Sigma 30mm f/1.4, which will set you back $400 or so.
A good starting point might be a good camera body (not necessarily a high-priced one) and a "standard" zoom lens to sort of get started with. Once you have familiarized yourself with the camera, you will have a better feel for what lens(es) to add next. Different camera manufacturers offer different zoom ranges and primes, plus there are a couple of decent third party lens makers (Sigma and Tamron) who make lenses to fit most camera models.
From a strictly Nikon perspective, a D40 plus 18-135mm lens is (IMO) a good starting point, although the D40, D40x and new D60 lack the autofocus motor necessary to autofocus some older Nikon lenses. Other camera manufacturers offer excellent cameras and lenses as well, so be sure and check out what's available from each.
Good luck!
~Y
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