There are regular buses linking Whitstable with Canterbury, Herne Bay and other nearby settlements.
Buses from Canterbury are particularly good, with several different routes (I used the 4A and the 6) running roughly every 15-20 minutes throughout the day (weekdays). Journey time is roughly 35 minutes. Some are double-decker buses, which give excellent views of the countryside if you sit upstairs.
You can find bus timetables and information on the link below.
Whitstable also has a railway station. But buses are generally cheaper than trains and, if you are travelling from Canterbury, they are also faster because Canterbury>Whitstable involves a change of train at Faversham. You can find train times, details and fares on www.nationalexpress.com
If you don't drive (like me) your best bet to get to Whitstable is by train. there is a good (half-hourly) service weekdays and journey time is around 90 minutes.
The Station is a little bit away from the main part of town, but here's a tip. if you come out of the Station door, turn hard right and then follow the cycle track signs. It takes you along some footpaths and is quicker than following the main roads.
The website will help you plan your journey.