Wales introduced the carrier bag charge in October 2011 - so we have had plenty of time to get used to it. Shops are required by law to charge at least 5p for every bag (plastic or paper) that you use. It applies to everything, even the brown paper bags that McDonald's put take-away items in. The shops are allowed to retain the charge for themselves if they want to, but most donate it to charity.
The big supermarkets are quite used to customers carrying re-usable bags from their competitors - nobody finds it strange now. If you buy clothes, the shops will always offer you a bag, you have the choice whether to have the bag and pay 5p for it, or take your goods without a bag.
When this was introduced in Wales it was heralded as being "something new", but in fact Germany, Austria, and some of the Scandinavian countries have had similar laws for much longer than Wales.