1) Get the right equipment.
When my son outgrew his baby carriage, we decided to buy a reasonably-priced umbrella stroller - not the cheapest, but certainly not top-of-the-line. The idea of spending hundreds of dollars on a stroller our child would only use for a few years seemed ridiculous. Less than a year later, we had run that poor little stroller into the ground and had to buy a replacement. This time we went with a Baby Jogger City Mini, which cost more than I would have imagined spending on a stroller... but had we just gotten a high-quality item in the first place, we would have saved ourselves the expense of the cheap stroller, as well as the hassle of finding a new one (on vacation, no less). Our City Mini still looks and feels new a year later, and I know we will be packing our kids around in it for years to come.
If your children are too young to walk the distances you will be covering, you will need a reliable and comfortable carrier, stroller, double stroller, or some combination thereof. My 2 1/2-year-old can walk further than most (and you can bet we're proud of that fact), but after about half a mile to a mile, walking with him becomes, shall we say, inefficient. You know the drill. Even as he gets older and more focused, I'm sure there will be times when we will tire him out with the distances we want to cover to get our errands done. Don't be embarrassed to be seen with your 4- or 5-year-old in a stroller if you're covering long distances (and make sure to have a stroller that will carry them comfortably).
If walking will be a part of your daily life with your children, allow yourself to splurge a bit on good-quality equipment. Think of this as an investment that will ultimately make it possible for you to save money by driving less. Please take note, I am not giving you free rein here to go out and buy a brand-new, thousand-dollar Bugaboo that will just sit and collect dust in your garage until you sell it on Craigslist in a few years (though if money is really no object, be my guest - they make some pretty amazing stuff). But I am giving you permission to look beyond the low-end umbrella strollers. Read the reviews - those cheap Disney strollers are for getting your kids from the minivan into the mall, not much more. Spending a bit more upfront will save you money, sanity, and health in the long run. If you enjoy using your stroller, you are also going to be more likely to use it more often.
2) Be prepared - but not too prepared.
If you're used to traveling by car with kids, you're likely in the habit of storing everything you could possibly need in the car - extra clothes, toys, diapers, snacks, shoes, hot and cold weather accessories - just to have your bases covered for any eventuality. After all, in the car, you have space for it, so why not? When you're walking, however, you will have less room and, really, less need for all the "just in case" gear. If you will be within walking distance of your home, most emergencies can be handled by just going home.
When we lived in the DC area, we always marveled at the parents and nannies who had strollers stuffed full of snacks, toys, and extra clothes. If our child got hungry playing on the playground, we went home for a snack. If he spilled something (rare because we didn't carry food with us) or got dirty, we took him home to change. Unless you will be out for the whole afternoon or day, only bring with you what you will need for your trip. And remember, even kids who get bored in the car will likely be entertained enough by the walk, nature, and your almost-undivided attention that they won't need toys or snacks to distract them.
3) Combine trips.
This seems like a no-brainer for parents, even those who drive everywhere, but it is surprising how often we give ourselves more trouble than we need to by not combining trips. If you are walking to the grocery store, is there anything you can pick up at the hardware store next door, or the library on the way? Does it make sense to go to a different grocery store that is a bit further away in order to stop in at other stores you might need?
This mindset makes sense for anyone doing errands without a car, but it is especially helpful for parents of young children. If you can work it out so that boring errands are interspersed with interesting or fun ones, or if you can squeeze in a trip to the park on the way home, then so much the better.
Beyond combining your own errands, is there any way to make your family's errands more efficient overall? Perhaps your spouse or a friend can pick up something for you on the way home from work, if it will be more convenient for them. If you need to make a purchase that will take some research, do the research you can online or even by phone first, rather than going to many different stores to see products in person.
One of the things I love about the car-free life is how this kind of thinking becomes second nature. Much less time is wasted driving around to different stores you don't really need, just because you can. It can make life with children much more pleasant because you also aren't dragging them to places unnecessarily, tiring everyone out in the process. Even errands that might normally be difficult with children become easier when the journey involves healthy exercise and fun interaction.