These are just some anecdotal tips based on experience with my 3 year old, so take this with a grain of salt.
If you want more professional advice on educational technology and very young children, research-wise, there is not much out there. The recommendations concerning young children and computers are mostly the same as with TV. Supervise your children, limit the amount of time they spend at the computer, don’t expose them to violent or inappropriate content. That includes games that involve shooting or exploding things. See this position statement by the National Association for the Education of Young Children for more info, although that statement is over ten years old (before there were online flash games, before Leapfrog Leapster and VTech V.Smile and similar toys).
Infants (> 6 months)
At this point your infant can sit in your lap and bang on the keyboard a while. There are keyboard bang games out there such as this online alphabet bang game from KiddieGames.com. This is basically a cause and effect activity, just like many electronic cause and effect toys we buy infants nowadays. They do some action, and the toy or screen responds with a sound and/or visual stimulus.
Use the F11 key to maximize your browser window when letting your infant bang on the keyboard. If you have a laptop, use an external keyboard if at all possible to prevent damage. If you let your infant bang on the keys while not in a game, be careful because they will find key combinations that you did not even know about.
Another idea is to open up a blank Word/OpenOffice document, change the font to something large and centered, maximimize the window, and let your child hit the keys to again see the causal relations.
Toddlers (> 1 year)
At this point my child was becoming more and more curious about the computer, started playing with the mouse, and learning that different keys did different things. My child learned from observing, too, eventually figuring out that pressing the start button turned the computer on (and much more later). At some point your child may be able to move the mouse and see how it is related to changes on-screen, but cannot yet point and click at objects on the screen.
You might start looking for software designed for toddlers that doesn’t involve point and clicking yet. Most technology and office supply stores have these kinds of CDs for very cheap. I highly recommend the Reader Rabbit Playtime for Baby and Toddler CD. It has games that gradually introduce mouse skills. At first you can just click or move the mouse to make things happen, and later you can click on objects on the screen.
At this point, too, you might want to create a separate account for your child, an account with limited access and privileges. Before you know it they will have figured out how to double-click icons to start a program or browser, how to move icons, how to turn on your computer, monitor, and speakers, how to click on their own name to login, and eventually even start a web browser and go to bookmarked sites.
Pre-Schoolers (3 and higher)
There are various games requiring different skill levels at those sites. Some of the games are very educational and will help your child learn things such as the alphabet, matching, turn-taking. Other games are not so educational but might help develop hand-eye coordination. I would recommend you model how to play the games for your child. Sometimes your child may not be capable of playing a game themselves yet, but they still can enjoy it through collaborating with you.
If you find any of these sites valuable, I would recommend you bookmark them and put them on the favorites bar of your child’s web browser so that they can be easily accessed. Your child will remember the games they liked to play, and eventually learn how to navigate the sites to find those games as well as try out new games.
There are many expensive software tools and electronic educational toys designed for pre-schoolers, such as LeapFrog’s Leapster and the V.Tech V.Smile. I tried the Leapster and it is very nice, but my 3 year old is not old enough for that yet. Also, frankly, the online flash games are much richer, more educational, and of course free. You can access hundreds of games at the sites linked to above, versus the hundreds of dollars it would cost to play more than a few games on the V.Tech or Leapfrog systems.
There has been research showing that the V.Tech/Leapster type electronic toys offer no significant advantage over traditional toys and learning experiences for young children, and I’d expect the same to be true of the online children’s flash games as well. It has helped supplement the activities we have done with our child, such as learning the alphabet and numbers.
One other item to consider for your child and the computer is videos and songs. If you have any videos or songs you would like to allow your child to watch, the free VLC Player can play any type of video or audio. After some modeling, your child can learn how to play/pause media, and how to use the scroll bar to jump to different sections of a video.
Your child will be curious about just about any technology you are using. Cell phones, PDAs, etc. My little one liked drawing on an iPAQ notepad, but be careful of items that are small or sharp, such as a PDA’s stylus. Keep potentially dangerous items out of reach of your child at all times.