How to use your camera:
I recommended getting your camera manual out today and figuring out what all those buttons are for and what your camera can do. Another option for some people might be a Field Guide. Many cameras have an entire book written about them. They are basically the same as a manual, but much more colorful and helpful. So, do a quick search on amazon to see if your camera might have a digital field guide.
Understanding Exposure by Bryan Peterson gives a lot more detailed explanation of the exposure triangle. If you are interested in understanding how to use the setting on your camera, then I highly recommend this book. And I have seen lots of other photographers that recommend it as well. (You can probably find it at the library for free.)
Anything by Scott Kelby is usually a fun and easy read. In The Digital Photography Book he explains how to do a lot of things that professionals do without being too technical. It is full of tips rather than an entire book about photography theory. Check out the product description on amazon to get a better idea (just click on the picture). But, I enjoyed it a lot and found it really helpful.
This website is a huge place for resources. It has tons of articles, tips and tutorials about photography, camera gear and post processing. It even has forums that allow you to post pictures and get feedback. It's great!
Flickr is a great resource for photographers. You can use it to find other photographers, get ideas/inspiration, and even constructive criticism. In order to get the most out of it though is to join groups. When you start commenting on other peoples' pictures, then you will start getting comments on yours. You can search for groups that have the same camera you have or the interests you have. For example, you can find a "Nikon D80" group and a "Children's Portraits" group.
Ok, so this one is awesome. Creative live provides lots of FREE classes on photography (and other things) if you watch them live. If you are unable to see them live, then you have to pay to view them. So, if your schedule permits you to, you will have the chance to view some really cool things. For example, next weekend, if you tune in, you get the chance to follow a professional children's photographer around to see how she runs her business from taking the pictures, interacting with the kids, to editing her pictures. It is a 3 day workshop. This may not interest you specifically, but you can look through their course catalog and see what is available.
Here are the softwares that I mentioned briefly today.
Picnik is a free website that allows you to upload a picture to their site and edit it there. It gives you a lot of basic options if you do not want to download any software to your computer. There is also an opportunity to upgrade for about $25 a year to access even more features.
If you have a mac, then you can easily use iPhoto. It is a great way to organize your pictures, do some basic edits and share easily through facebook, emails, twitter, etc.
This is what I used for a long time. It is a free software from google. You can do a lot of stuff on it like straighten pictures, remove red eyes, make collages, add words and even do some retouching (to remove food from your child's face). It is also very easy to share your pictures from this program.
I really do not know anything about Gimp. But, I have heard that it has similar features to photoshop. And may give you a little more control over your editing. So, like I said, I don't know anything about it, but you might want to check it out. (sorry that wasn't very helpful)
Photoshop can do amazing things. But, it is probably way more than you need for just editing your family pictures. But, if you are wanting to get professional looking pictures, then you will need photoshop. Look for the student discount or even photoshop elements which are both much cheaper.
If you have photoshop, then you know that it isn't an easy piece of software to just sit down and figure out. So, I would recommend this book by Scott Kelby, The Adobe Photoshop CS4 Book. (Or if you have a different version, you should get the one for your version.) Once again, it is by Scott Kelby, so it is easy reading. He gives a lot of step by step tutorials to help you do things in Photoshop. I still use it as a great reference book.