4 years ago I was in your shoes, wanting to take better pictures (of my kids primarily). I received a Nikon D40 as a gift, and have gone on from there to using all manner of cameras

There are lots and lots of tradeoffs to consider, unfortunately. Generally speaking the SLRs will set up, focus, and click much much faster than anything point and shoot. Larger sensors perform better, especially in Low light. See this

https://en.wikipedia.org/wiki/Image_sensor_format

You don't need the latest generation of DSLR to get started, and I don't really recommend that you start with a lot of features. There are a lot of things to learn if you get into it. I know you want to keep it simple, and every _photographer_ (as opposed to gearhead) wants their stuff to get out of the way so they can take pictures. Unfortunately there are a lot of decisions to be made, gear and otherwise with each click and making better decisions with more capable gear means better pictures. Pro pictures look pro for a reason.

Having enough gear to make those decisions is important, but learning what they are is more important. Things like composition and lighting, and how to bounce-flash when you can get away with it.

You can always buy gear on CL and sell it the same way when you have exhausted its limits. You can more or less try gear out "for free" with a deposit that way. Especially if you stick with "popular" (hence semi-liquid) brands/items.

With all that said, buy either a used D40 or D3100 off of Craigslist with a kit lens (or the canon equivalent), or get an Olympus E-P3, E-PL3, E-P1 or Panasonic GF3 (with kit lens). The new m43 bodies set up and shoot much much faster than previous "small" cameras and the m43 sensor is big enough to be "good". And they are a heck of a lot less bulky than the SLRs.

Fantastic deals on those today

http://www.43rumors.com/black-friday-brings-superdeals-on-e-pl3-e-p3-e-5-and-gf3/