Both British and American speakers can refer to everything that travellers carry as their bags. American speakers can also call any individual suitcase or similar item a bag.

1. ‘Let me carry those for you,’ he said as I picked up the bags.
2. She got out on the sidewalk and the porter took her bags.
3. It was a fact that an airline passenger’s bag could not be counted on to arrive safely at Pine Bluff, Arkansas.

Speakers of British English normally use luggage when they are talking about everything that travellers carry. However, they sometimes use baggage when they are talking about travelling in a technical way, for example when discussing airports or travel insurance.

4. There are also coin-operated lockers in which you may leave luggage.
5. If you lose any of your luggage, enquire at once at the Lost Property Office.
6. Your baggage, clothing and personal effects are covered up to one thousand pounds.

In American English, luggage refers to empty bags and suitcases. Baggage refers to bags and suitcases with their contents.

7. He pulls in behind a Citroen with a Paris license, takes out his baggage, and locks the car.

Note that there is no plural form of baggage or luggage.