I check my receipts always. If they make a mistake I call them out on the spot and insist on getting the lowest posted price I'd seen that's current, usually being what it says in front of the item. I've not given thought that they'd intentionally cheat you by having different prices, one where the item is on display and the other in their system. I figure they goofed up. Doesn't matter from my perspective. It was a lapse, they are responsible, they were derelict, I want the price I expected. It doesn't happen a lot. I usually know what I'm paying.
Now, people with enough money don't do those things, at least a lot of them. They toss things in their cart and don't pay attention to the price tag. That's NOT me.
I go a lot further than most people. For food items at least I keep data. I enter it from receipt when I get home. I can look up and tell you what I last paid for potatoes, coffee, butter, all that stuff, where I got it, the date. What the total was for the receipt. I sometimes add additional info for my possible edification at a later time.
My online purchases, I keep info on all that too. Sometimes not as thoroughly as I would like when I visit it later, but I make an effort to link the listing (whether Amazon, ebay, or whereever), the prices for items, dates, other pertinent info. I have found those things very useful many many times.