I take all my own bags with me and have done for around 25 years now. I have around 20 assorted sizes and styles, including cool bags for cold and frozen goods. If you are combining food shopping with other shopping then make use of the stores storage section for keeping your bags until you are ready to enter the supermarket. I also try to cut down on the plastic for my fruit & veg by multi bagging items and getting labels put on larger items i.e peppers, aubergines etc.. I always have a fold up bag in my handbag for those odd items I need to pick up.
in answer to the above comment, charging for plastic bags is the only way to make people who don't care about the environment notice there is a cost to producing and overusing plastic!