I go to the same grocery store now as I did before I went vegan. I feel like the best way to save money is to buy fruit and veg that are in season and to only use heavily processed meat substitutes as a treat instead of an everyday thing. Beans, lentils, and even tofu are pretty cheap.
Also, if you have the resources and space, growing veg and herbs from seed can save money and make meals tastier.
Trader Joe’s has good cheap vegan stuff
beans and rice are still pretty cheap even for premium.
it’s the premade food that’s expensive to match with takeout in your region.
In terms of big stores, Wegman’s has a ton of plant-based products as their store-brand option. They mark whether or not it’s vegan as part of their packaging scheme so it makes for an easy trip.