The simplest approach is by train from Paris Saint-Lazare to Trouville-Deauville station — a direct service that runs around two hours and lands you a five-minute walk from both town centres. Several departures a day in summer, fewer in winter; SNCF often runs themed promotional fares to the seaside.
By car, Deauville is about two and a half hours from central Paris on the A13 motorway via Pont-l'Evêque, longer in summer holiday traffic. The small Deauville–Saint-Gatien airport (DOL) handles seasonal flights from London City and a handful of other European cities; otherwise the larger Caen-Carpiquet airport is an hour's drive west and Paris Charles de Gaulle is two and a half hours by direct shuttle bus.