eBuyer is a good website for buying electrical stuff (www.ebuyer.co.uk) - I have used them in the past and they have generally delivered the goods, so to speak. There are a number of useful tools so you can track the status of your order... if you enjoy that kind of thing!
Anyway, they have a good selection of cameras, and useful reviews by people who have bought them already.
My choice would be Canon - the IXUS range if you want a tiny camera, or the Powershot range if you want AA batteries, a viewfinder, and generally more features for your money. Optical zooms are better than digital zooms as they physically move the lens, rather than simply blowing up the image (with consequent loss of resolution) which is what digital zooms do, but generally speaking, the smaller the camera, the harder it is to fit an optical zoom in - so expect to pay for it. Ideally you'd want a camera that has automatic controls (for when you're feeling lazy or your subject is running away) but also the option to control settings manually (for when you need more control, or want to experiment). Viewfinders tend to disappear the smaller the camera gets, but they're useful features as they save having to use the LCD screen to compose your shots (which is difficult in bright sunlight, and drains your batteries).
Also, factor in the additional cost of extra memory cards (eBuyer do these much cheaper than you'd find on the High Street), a case, and spare batteries (and a charger).
It's probably worth nipping to WHSmith and having a quick read of a Digital Camera magazine to get an idea of what the experts say and which cameras stand out from the hundreds available...
I've just realised how much I've written...