When I was researching photography equipment before I bought my camera, I found the Pioneer Woman's photography site very helpful. In all her posts she assumes the reader is a beginner. I found this post extremely helpful. In it she explains all her photography equipment, the camera she uses and what different types of lenses there are and how to use them. It's a great introduction.
I have a Nikon D80. They actually discontinued it right after I bought it. Similar cameras are the D5000 or the D90. The D90 & D5000 have the ability to take high definition video so they are more expensive than my D80 was. Another, cheaper option without video is the D3000. Canon and Nikon are the best brands on the market and you really can't go wrong with either one (my brother and sister both have Canons and they are very similar to my Nikon). I believe Sony also has some good cameras that are a little cheaper, but I don't have any experience with those.
A great place to buy a camera online is B&H. They have excellent customer service (the people answering the phones are actual photographers and know what they are talking about). You may be able to find cameras cheaper at Amazon. Even if you buy elsewhere, I suggest checking out B&H because there is so much product information on their website and it will help you figure out what camera is best for you.
I currently have two lenses: the 18-135mm zoom lens that came with my camera and a 50mm lens. The zoom lens is great when you just want to "capture the moment". For instance, a birthday party or special event where you just want to document the event, not necessarily create amazing photos (not that you can't create amazing photos with it, too!). The versatility of a zoom lens is essential, especially when you're trying to photograph children that are always on the move. My 50mm is fantastic for taking portraits (my latest picture of Clara was taken with my 50mm), for indoor photography without using flash (it has a very large aperture that lets in lots of light) and for pictures where you want a very shallow depth of field (when the subject is in focus but the background is all blurry. My butterfly picture is an example of a shallow depth of field shot taken with my 50mm.) The 50mm is a relatively cheap but also very high quality lens and I highly recommend it.
The Pioneer Woman has a series of posts about the basics of photography that are very good. Check out her "what the heck" series of posts that teaches the basics of exposure. Another good resource is The Digital Photography book by Scott Kelby and Understanding Exposure by Bryan Peterson. I found both books at my local library but eventually I bought a copy of each for reference. Another more advanced resource I use is the blog, Digital Photography School. It's a great resource for post-processing tutorials, info on the latest photography equipment and different tips and techniques.
I don't know if you are interested in post-processing or not, but I photoshop all of my pictures. I use the cheap version of photoshop, Photoshop Elements which is less than $80 on amazon (the professional version of Photoshop is more than $600!). The Pioneer Woman has a bunch of photoshop tutorials and I also use Scott Kelby's The Photoshop Elements Book for Digital Photographers. I have to say, I love post-processing just as much as the actual picture taking itself. I think you can create very professional looking images with Photoshop Elements.
I have also learned so much from posting my pictures and viewing other photographers pictures on flickr. I've picked up so much about composition and post-processing by studying the work of other great photographers. Some of my favorite photographers are Kiwi Gal, Shana Rae, Lisa at TSS, Mike Golding, and Sarah J Gardner.
