Over the course of our travels we have always found that one of the best ways to get to know a particular location is to shop with the locals. This can vary between going to a big supermarket, visiting a convenience store, or buying something from a non-tourist market.
Supermarkets tend to be a really good way to get a grasp on what living in a particular location is. You are able to see how prices vary between your home location and your holiday location for common products. You can view and try unique items and services which are not available at home and you can save yourself some money on things like drinks and snacks and maybe even entire meals if you are so inclined. We have had meals from supermarkets in Mexico, Portugal, Spain, Hawaii, Florida and many more locations. Almost all of them have been tasty and easy with some being real standouts. I would never recommend that you stop eating out since this is something that we love doing also. It does however give you more options if you want to have a lazy evening.
One of the best buys in our experience tends to be large cartons of fruit juice since buying drinks while on the go over the course of a two to three week holiday can really add up. Supermarkets are also great for buying mass produced souvenirs since they tend to have a much smaller markup compared to tourist shops. For quality souvenirs I would always suggest an authentic local craftsman but supermarkets are great for t-shirts, magnets and keyrings.
Another big advantage of supermarkets and convenience stores is if you want to have some alcohol in your hotel room in the evening. Buying bottles of wine, spirits and mixers or pre-mixed cocktails and bringing them back to your room can save a considerable amount of money whilst also increasing your range of options.
Lastly, if there are public or hotel cooking appliances available, it can be a great cost saving measure to do some self cooking even if you are not on a self catering holiday. We have for example used the free public grills on Cairns Esplanade to cook a full evening meal for considerably less money than it would have cost to visit a restaurant (we checked). Plus you get an interesting story to tell afterwards.
Preparing our Surf and Turf on Cairns Esplanade 
About to Chow Down
All in all I am a big supermarket/convenience store fan, it gives you the opportunity to see things like the insane range of hotdogs that they sell in 7-Eleven in the Philippines or the incredibly impressive tortilla making apparatus that runs in supermarkets in Yucatán, Mexico. I highly advise you drop in.
