You might want to dive into couponing headfirst, but easing in is probably the better route.
Start with a single store while you're getting your feet wet; you can branch out later as you get used to bargain pricing and coupon policies. Find a couple of favorite cashiers to frequent who don't mind the drill. Make sure to thank those cashiers and compliment them often. Warn people who get behind you in line that they may want to choose another aisle. Smile whenever a situation seems like it's starting to sour.
The point is that it's important to remember that while you're learning an exciting and cost-saving new practice, other people are just trying to pick up a few things for dinner after a long day at work. Which reminds us of another tip: Avoid shopping during peak hours and trips to the store will tend to go a lot more smoothly.