Digital photography is great! You can take hundreds of photos on one memory card and store them easily on your computer. A common issue I run into with some people is they misunderstand the longevity of digital photographs. They believe that these photographs will be safely stored forever on their computer. The truth is print photographs have a longer life span than digital photographs and that has to do with a technology term called "digital migration".
Digital migration is moving your digital photographs from one electronic device to another. For example you create digital photograph with your phone/scanner/camera and then usually transfer it to a computer. The digital migration is not over for that photograph yet, because you will probably have to move it to several other computers in your lifetime. Consumer Reports suggests replacing your computer every 4 to 6 years. If you move your images on to each new computer, suddenly by the time your new baby gets old enough to graduate high school you have "migrated" 3 or 4 times. Every time you complete that process the chances of those images getting lost or damaged is increased. To avoid those problems I suggest saving them to a online web album.
You can utilize a online web album for easier access and saving of your photos. There are a ton of free services and I am a big fan of Picasa. I think this Google provided program is awesome because is easy to download, will search your computer for all of your photographs, comes with a simple free photo editing software, and gives a lot of control over if anyone can view your photos on the internet. You can also clearly label each photo, change the size of the photo, and correct some of the errors like "red eye". If you find Picasa not to your liking there are other free services like Photobucket and Flickr.
Once you download Picasa and start uploading your photographs you save your computers in a "online web album". That way if your computer ever blows up, melts, freezes, or gets stolen, your photos will be saved and accessible through any other computer with internet access. You can also save your photos in different albums. Here is an example of what a album with one photo would look like (that is my nephew..isn't he cute).
If you are interested in getting started with Picasa I suggest checking out some of the intro material. If you are still a little confused stop in at the library and the staff can assist you in understanding. Good luck with your photos!