Specifically what sort of budget are we talking about?
The camera body is a foundation to the system with a DSLR, but the lens prices for any camera can get pretty high.
The manufacturer's tend to create a few lenses with budget in mind ... keeping prices in the $100-200 range (perhaps even up to $300). But much of the lenses can be $500 or more and there while they do have lenses in excess of $10k (even if few people can afford these) there are a lot of lenses in the $2000 range that are extremely popular.
HOWEVER... in addition to Nikon or Canon glass, there are a number of 3rd parties that make glass to fit Nikon and Canon bodies (e.g. Sigma, Tokina, Tamron, etc.) and these are generally less expensive (often significantly less expensive.)
As budget is a concern, go directly to Nikon's online store as well as Canon's online store and check the "refurbished" sections. You can often save a significant amount by purchasing a refurbished body and the manufacturers provide a warranty with these (Canon's refurbished warranty is the same as the new warranty. I don't know the nuances of Nikon's warranty to know if their refurbished warranty is also the same.)
For Nikon, the D3xxx series are their base entry-level bodies, the D5xxx series are the still considered entry-level... but at the high end of the entry levels. The 7xxx are mid-range (not quite "pro" but better than "entry" range).
For Canon, the T3 and T5 (without the "i" suffix) are the base entry-level bodies. The T3i and T5i (with the "i" suffix) are the high end of the entry-range. The 2 digit bodies (60D and 70D) are mid-range bodies. Single digit bodies are higher end (and of course noticeably more expensive.)
As for the NEX-C3 vs. Canon and Nikon images... the "sharpness" is largely a function of the lens... not the sensor so much (there is one nuance where it can be). But mostly what we look at is a comparison of how much "noise" is present in the image as we increase the ISO. As I scrutinize the samples at DPreview, it looks like the Nikon and Canon bodies I'm comparing to (Nikon D5100 & Canon T3i) have a very slight edge on the NEX-C3. It's not a huge difference though, but I think they do have slightly less noise.
As for performance in very low light (night shoots), you'll find full-frame bodies have a rather noticeable difference (improved) over crop-frame sensor bodies. E.g. something like a Nikon D610 or a Canon 6D. But the price jump for those full-frame cameras is fairly significant (and those are the "entry" level full-frame cameras.) Given that you don't own any lenses and would need to buy a camera body and at least one lens, the price for either of these is north of $2k.