It is hard to find the time and motivation to start a garden - or continue a garden you had in your pre-parenting life - with kids around. But, as you probably realize if you're reading this, it is worth it. The time spent together, time spent outside, learning opportunities, and unspoken metaphors start paying off very quickly. Plus, you may get a few berries out of it (if your kids don't beat you to the harvest ... which they will).
You don't need special dirt! You don't need a lot of space! You don't even need special trips to Home Depot and the like (although I have reviewed some local nurseries for you) - just pick up some seeds and pots at the Dollar Store, where you know you were going anyway for the bubbles and crayons.
Get going with these posts on some of the basics.
Planning Your Garden, Part 1 and Part 2 (so square, I know, but after a year of trial and error, you'll want to do it too. You might even be more organized and less Undergardener than me, and do it from the beginning!)
Gear You Need
Getting the Kids Excited
Keeping Them Busy