Public transportation
It will surprise many people, but there is no public transportation in Deauville. The resort is so small that all the locals walk or drive their own cars. Tourist tip: if you need transportation to get around Deauville, but you can't rent a car, ask about hotel buses. For example, the Royal and Normandy offer free shuttles to the racetrack or casino.
Cost
The only more or less "public" transportation in Deauville is the cab. You can order a car by phone: +33-231-98-51-66, +33-231-87-15-95. The price of this transportation in Deauville is 10-20 EUR per trip. Plus a tip (optional).
But intercity transportation in Deauville is very good. You can take buses from the city to almost anywhere in Lower Normandy, as well as Paris. For more information on routes, check here.