Travelling in Thailand is easy - From Bangkok you can take an airplane to almost every popular tourist destination like Phuket and Phi Phi Island.
Going from Bangkok to Pattaya is normal with bus or taxi/minibus and the journey take 2-2,5 hours.
Thailand have railroads, but it seems like its more popular to travel with bus and you can get almost everywhere.
Booking air tickets or other tickets can be done in one of the many travel agents shops - and many hotels have their own travelling agent who can help book a trip and confirm and order tickets - or perhaps rent a car.
If you rent a car make sure everything is OK about insurance.