We have two kids- age 4 and age 1 and currently don't have a vehicle. For the older child, either myself or the DW will ride bikes with her to school. For the 1 year old we just have him in our stroller. For trips to the store we just walk with the stroller, which can carry our groceries, plus i'll have on a back pack. Same thing with trips to the library or the park. We'll also take buses, and there is a free neighborhood circulator that you can catch every hour that goes places.
For out of town trips, or trips that aren't convenient for a bus, we'll generally rent a car either for a day, or weekend. Enterprise has had a great deal for weekends the last few months-$9.99 a day for a compact car. One thing we've learned going carless is that we can definitely make a smaller car work (we don't need the armor plated SUV) so when we do go shopping for a car we know that we can go for something smaller sized and affordable.
I would also agree that its much easier when the child is smaller. Now that our daughter is 4 she's got more friends, so more parents want to do play dates, birth day parties, etc. Our daughter wanted to go to the zoo the other day... things like that. Not everything ends up being on a bus route.
I've also been able to pretty much exclusively work from home (and DW is SAHM) so that helps a lot. Although, it looks like i'm going to have to start working across town a couple of days a week, and with Kindergarten approaching, it looks like we're going to have to break down and buy some sort of vehicle. But i've enjoyed our car free existence! It's great not having to worry about maintenance, insurance, etc..