Yes, you can safely go grocery shopping, but the best way to protect yourself and others is to limit exposure to the grocery store. This can be done by ordering delivery if that is possible, or reducing the number of shopping trips you take. Try to only go to the store every one or two weeks. Also try to limit the number of people who go on shopping trips. Leave your kids at home if you can as they like to touch surfaces and can carry the virus without showing symptoms.
Other safe practices are wiping down the handles of your cart with disinfectant wipes before use and leaving your phone in your pocket while in the store. Try to go back to a paper shopping list. If you bring reusable bags to the store you should wash them either with wipes or in the washing machine before using them again.
To pay, it is safest to use contactless pay, like Apple Pay. Credit cards and cash are safe as long as you are careful not to touch your face, and to disinfect your hands after you touch the pinpad.
The risk of going shopping is not the foods or packaging, but the people you encounter. However, you should always wash your hands before and after handling food. The CDC also recommends you wash your hands again after you unload your groceries, and clean kitchen surfaces like countertops, cabinet handles and light switches.
It is not necessary to sanitize your groceries or leave them outside after purchasing them. The FDA and CDC report that there is no evidence of human or animal food or food packaging being associated with transmission of the coronavirus that causes COVID-19. Wash your fruits and vegetables like you would under normal circumstances.
The CDC recommends wearing cloth face coverings in public settings where other social distancing measures are difficult to maintain (e.g., grocery stores and pharmacies). It is also essential to practice social distancing and stand six feet away from everyone in the store. Try not to touch your face, use hand sanitizer and wash your hands as soon as possible.