You can always ask the business you're visiting if you're allowed to bring your dog along while you do your shopping or get a bite to eat. You may be surprised to learn how many businesses are happy to allow dogs inside, as long as they're on a leash or in a carrier. Due to health codes, not all businesses are legally able to do this, but it never hurts to ask.
Some businesses are even incorporating temperature-controlled dog kennels in front of their stores. This allows you to pay to put your dog in the kennel, safely locked, while you do your shopping. When you're done, you can let your dog out, pay, and head on your way.
If you're driving with someone else, they could wait with your dog — as long as they're on a leash — in a shady spot outside of the car while you do what you need to do inside.
The safest option, however, is to leave your dog at home (or drop them off at a boarding kennel), especially if you'll be making stops at places where they're not allowed inside.