Just to throw another little bug in your ear when shopping...I'm surprised to see one of the big manufacturers left out of the suggestions so far. Canon, Nikon, Sony, and Pentax are all great...and Olympus is quite fine too. The choices don't stop there, as there are DSLRs out there by a few other manufacturers and they too will do a fine job. But these 'big five' are probably the best selling, most established, and have the largest lens selections, so they tend to be the ones most commonly recommended and for good reason.
If you want small and easier to handle - you can't go too wrong with the new Sonys mentioned -they do seem to have been designed quite specifically with P&S converts in mind...being smaller and with much more walk-through menus and descriptions. Also consider Olympus, who has usually made some of the smallest and lightest DSLRs on the market. They use a slightly smaller sensor than the other entry level cameras, but don't seem to suffer too much from it at all - in tests they rank right up with the others in image quality. And their Zuiko lenses are some of the better lenses available for entry levels.
As also mentioned, how and what you intend to shoot, how willing you are to really get into photography, how much money you want to spend - all will be factors for what to recommend. Truth is, a heavy majority of all amateur photogs out there would be served just fine by a P&S camera...with those more enthusiastic about photography getting great results off of an advanced P&S with some manual controls. Many newbies go straight to a DSLR, but don't necessarily need it, or use it to its abilities. And that's fine - DSLRs are the best cameras for image quality in the widest range of photographic situations...but just think it through and decide if you want to accept the compromises you'll have to take to go to a DSLR (hey...there are ALWAYS compromises!). With a P&S, you get small, portable, sealed, cheap, easy...but you don't get as good with action, movement freezing, high ISO, or ability to change lenses. With a DSLR, you get flexibility, lens range, high ISO, fast speed, tracking AF, optical viewfinders...but you don't get small, light, sealed or cheap...and easy is debatable.
I shot for years with advanced P&S cameras, and got tremendous results, was very happy, and managed to sell and publish photos from those cameras. It took 3-5 years of learning and evolving in photography before I was truly up against the limit of what I could achieve with those cameras...and had to move to a DSLR. Believe me, a good P&S can be all some folks ever need - and you can actually make money from photography using one and get good enough results for large prints that look professional and national publication standards to be met. I shot wildlife, landscape, night scenes, events, and portraits with P&S cameras, and made enough to buy all my DSLR equipment. I needed to move up into higher ISOs, shoot more moving subjects and action, needed faster burst modes, low light lenses, super-telephotos, and ultra-wides. Only when I wanted to branch into that type of photography did I really NEED to move to a DSLR. And even then, I couldn't get by with P mode and a kit lens - I had to spend thousands on specialty lenses to suit those specific tasks. DSLRs can be a very pricey habit, and lens lust is a real disease that can drain a wallet - so beware! Make sure you REALLY feel you need a DSLR, or want one really really badly, before making the final decision. Or next thing you know, you'll have 10 lenses and a hand-cart to travel with them all!
Will look into the book!
At miniature dachsunds, do you think a hyper active 3 year old cat would do the same job? 
