Having your child find or print coupons for items on their list isn't a difficult task for most middle-aged to older children. If you already coupon, they might search through your current coupons for ones they need. Online searches for products via search engines or websites like coupons.com can help them in their search. Of course, Sunday papers provide coupons, but some Sundays in the month are better than others. It won't always prove beneficial, but giving them a little exposure to planning, saving and searching certainly will help them later in life. Also, show them that a coupon won't always save you money (i.e. if you can just buy the store brand for less money), but it can be like cash when used for items you were already going to purchase.
First, explain to your child that (generally) it isn't cost effective or good time management to hit several stores just to save some money. Next, explore the store circulars to decide if one particular local grocery store might be better for what you need than another. Let's say they are making a dish that requires ground beef and ground beef is on sale at the X grocery store but not at the Y grocery store. Meat is often the biggest expense and following the store ads is typically the only way to save on it, so perhaps getting all your items at the X grocery store would be the wisest choice. Also show them how coupons and sales can combine to make pretty significant savings.
You may consider giving an older child a budget for the meal and shopping trip. This may prove frustrating for some, but for children close to graduation, it really is a great dose in reality. Learning how to make a dollar stretch and stay within your means, is a lesson that all future adults (and many current adults) need to learn.
This may take a lot more time for a newbie, but eventually when one gains knowledge and associates a "usual cost" for something, it takes much less time. For each item they need to try and find the best deal. This is a way to show them that the lowest price isn't always the best deal. An item may be in a smaller package for only a little less money or a package that is twice the size of the smaller one that costs more than double the alternative. Many stores show you cost per oz. in the small corner of the price tags. This is the only effective way to compare the cost of items. If the store doesn't have this, show them how to calculate it (use your phone calculator or if you happen to bring one, it can be handy). I wouldn't exhaust them on this, because they may never do it again as an adult, but exposing them to the ideas can give them awareness needed for the future.
Sticking to the List
Last of all, insist they stick to the list. Have them carefully follow and check off the list. They can use notes to remember if they have coupons for items.