I have found that always having everything you need in the car works wonders when it comes to getting out the door on time and keeping your child happy while on the go. My wife and I have the essentials in the car at all times. Then when it comes time to get ready, things are a lot smoother.
Here's what you need:
Toys, toys and lots of toys
Diapers & wipes
A change of clothes
Now that you have what you need in your car it's time to get ready. Start by doubling the time it took you regularly to get out the door. This will help make things easier and hopefully headache free. In a two person household you can have one person get ready peacefully while the other watches the kids and then vice versa. In a single person home, you'll need the extra time chasing your kid(s) around while you get ready yourself first and then them. If you're lucky, find something that will keep your child busy and out of trouble while you both get ready!
In the car, keeping your child busy is key. When he was younger, my wife used to sit in the backseat to entertain our son with his toys. Now that he's over two years old, he can entertain himself with his toys and he also LOVES telling us what songs he does and doesn't like on the radio.
When it comes to the grocery store or other businesses, my wife has a great system worked out! Find something at each business you often frequent that your child loves and will look forward to each time you go there. For example, most grocery stores offer free cookies to children. At the end of a successful trip up and down the aisles you can reward them with a cookie of their choice! At Target, most have cafes where you can buy popcorn instead. Others have Starbucks inside where our son LOVES tea or apple juice as his reward. He sees these stores as we drive by and begs to go inside for a cookie or popcorn knowing he has to be well-behaved the entire time.
In the car, aside from toys, keeping your "car essential kit" helps in case of emergencies like potty accidents, food stains, tears on clothes, etc. You'll always have a stroller in the truck because you never know when you might end up having to walk somewhere far with your child and this also works for a "nap-time on the go" alternative.
These aren't all the items you'll need though. You might need more or less but we've found always having these things in the car to start with cuts down on having to remember all these things every time we're trying to get out of the house quickly. It's always better to be prepared!