i.e. Joe Sixpack is going to pay 7% on his loaf of bread, but Jane Wicstamps buying the same loaf of bread with food stamps does not get charged any tax.
Additionally, some municipalities will do things like say there's a general 7% sales tax on everything, except groceries, which are taxed at 2%...until next year, when the general tax goes up to 8% and the tag on literally every item in the store needs to have the price adjusted.
For things that don’t have different VAT rates, such as food or children’s clothing, it’s the retailers responsibility to price that in - most people probably couldn’t even tell you which particular goods are VAT exempt.
Computers can deal with the latter, signage can't.
Might be a little different for online stores where tax can't be calculated until they know which of the many different local tax codes the sale is being made under, but as you've already noted computers can handle updating the price before people confirm the purchase.
Except for businesses, there isn't the situation as in the US, where the retailer is responsible for verifying that a customer is tax-exempt. People on the equivalents of food stamps pay the VAT; yes the money just goes in a circle back to the government, but it's probably easier accounting for everyone.
The till receipt or invoice breaks down the VAT element for you, and most places even pick out what is zero rated or exempt should you care.
Every receipt breaks out the VAT and rate. Food, books and children's clothes (and a few other things) are exempt (UK).
All online purchases show VAT in the checkout and basket, and in the invoice after purchase.