There's nowt wrong with second hand. Both my DSLRs have been second hand and both have been well below your limit.
Also there other brands than just Nikon and Canon, although they are the most popular for DSLRs.
I have a Sony DSLR and it's really good. BTW Sony make the sensors (the image capturing bit) for Nikon's DSLRs (except the very top models), so they are certainly not lower quality.
Sonys are cheaper second hand because they're less desirable, but still very good.
As suggested though, go into a shop like Jessops and try out a few. With DSLRs in particular it's about how they feel in your hands, how easy it is for you to change settings (like ISO, shutter speed, aperture/f-stop), etc.
If you were used to a manual SLR, then you should remember about the exposure triangle and how to lower the f-stop to compensate for a faster shutter speed, etc. The difference with digital is that you can also change the ISO (light sensitivity) of the image sensor, unlike film which once you loaded a 400 ISO film, you were stuck with it.
Here's the cheapest model from each of the three main manufacturers, each comes with the basic lens, but you'd still need to get a memory card and a case:
Canon 1000D £350 - just been superceded by the 1100D so you could find some good deals on this.
Nikon D3100 £400 (after cashback from Nikon) - will record video as well.
Sony Alpha a290 £290
Once you've bought a DSLR body from a certain maker, you can only use lenses made for that system, so you can't swap lenses from a Nikon to a Canon and so on. Canon, Nikon and Sony all make their own lenses, but there are also 3rd party companies that make lenses for all the cameras. The most popular of these brands are Sigma and Tamron. These tend to be slightly cheaper to buy than the camera manufacturer's lenses.
If buying new from somewhere like Jessops, they'll often do a "bundle" deal on a camera with an extra lens and accessories. But you can easily just get the basics and learn with that first, then buy more later.
If you're taking photos of kids you may want to leave the camera in auto-focus because they move quicker than you can focus the camera, but with all DSLRs you can easily switch from manual to auto-focus. Most budget DSLRs also have a number of auto exposure settings, called "scenes" so you can put it onto "sports" for a fast shutter speed, or "landscape" for optimal landscape shots and let the camera do all the work. Or you can put it into other modes that give you varying degrees of control over shutter speed, aperture or both.
