Play a color-based game with your child. By the age of 2 or 3, most kids know how to use a mouse, but knowing how to use computer hardware isn't really essential. Visit Do To Learn and check out the "What Color" game, a flash-animated game that will ask your child (or you) to click on the correct color for a series of questions related to the Gumby-like character "Bud." It's very basic stuff, but features funny animations that will engage kids.
Story Place has another interactive, color learning game called "What Color Is It?," which features an animated frog and hippo asking trivia questions about colors. The voice acting for a simple flash game is excellent.
Play shape games with your child. The PBS Kids website has a wide selection of kids' games based on characters from the PBS lineup. Your kids can play matching and shape-finding games with Caillou or Jay Jay the Jetplane. Both of these games are a bit more advanced, seeking to match complex shapes, but still teaching valuable skills. These are more appropriate for first- or second-grade students, as the games teach about relative shapes and shape transposition.
For the basics, check the Kids Online Resource, which features an early-childhood game for learning squares, triangles and circles. Each shape has printable worksheets for coloring.
Read fun, interactive stories with your child. Web pages don't need to be geared specifically toward learning shapes and colors for you to teach your child about these subjects. After you've worked out the basics by using the games in steps one and two, read through some of the interactive stories on Starfall, a mostly free website devoted to educational activities. The readers are sorted into four categories, from learning to read phonics activities to plays and comics, but each level features animations and sounds that keep kids interested. Read through the stories and ask questions about the characters to review, like "What color is Zac the Rat's shirt?" and "What shape is the house?"
Work with printables. Most of the websites mentioned here allow you to print components of the learning pages for coloring. Building a pattern with your child can help reinforce the lessons from the games. Work through an activity or read a story with them, then print off a few of the pages from it and sit down and color together. Ask questions to recall the story, like "Can you remember what color her hat was?" and then color the pages accordingly. Or let your child use their imagination, then ask questions about the page after.