We do both. It pretty much depends where we are at the time. However, Walmart has other things, so we usually schedule a stop there. Make sure you go with a list. It's very easy to buy too much and then have problems at the end of the trip, trying to pack.
We shop mostly at Publix ... convenience as its right opposite our hotel. But we do go to walmart for some stuff as well. If you have a Costco card, there is great deals there too.