Before I bought the £100 Panasonic Lumix, I went up and down the budget scale. If I spend a hundred quid more, I could get more gizmology? I was almost up to £500 before I realized that technology creep was getting a bit out of hand. Just get a camera that is easy to carry. £100 in my case.
I decided on point-and shoot. I never had ambitions to challenge Tim Page. If you have an eye for a picture (and I certainly don't), the camera is pretty irrelevant.
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If all you want to do is family snaps, landscapes and touristy type stuff and have a camera that you can easily slip into a pocket, then a P&S is perfectly fine. Actually, hell, the camera on a smartphone will do that job reasonably well.
But given that the OP wants to do some sports photography, it is pretty well much a given that this will exceed the capabilities of a P&S or smartphone and quickly lead to frustration. I know. I've been there. For sports photography you really do need something with a good reliable AF that locks onto your subject quickly, plus a reasonable frame rate - say, 5 fps as a minimum, and, more importantly, the ability to shoot in shutter priority mode (useful in bad light) or better still, full manual.
It's not so much about "gizmology", it's about getting the right camera for the job in hand - you wouldn't undertake a DIY or car repair project without the right tools and equipment. As you go further up the range of DSLRs, you actually get progressively fewer fancy bells and whistles and pre-programmed modes than on the bottom-of-the-range jobbies - what you do get are bigger sensors, better AF with more focus points, higher frame rates, a bigger buffer, better battery packs, and the flexibility of being able to set up the camera to suit your needs.
A mid-range DSLR will offer the capability the OP wants for sports photography and yet still not be so big as to be unwieldy. If one can live without having the latest model, there are some good deals on the secondhand market.
I saw a very tidy Canon 40D body in CEX previous week for £90...