Vehicular Ad-hoc Network (VANET) services use a range of information, such as traffic conditions and location information, for safe and convenient driving. Information exchange in VANET happens in Vehicle-to-Infrastructure (V2I) and Vehicle-to-Vehicle (V2V) communication modes. Several V2I and V2V authentication protocol suites are proposed to protect the information from attacks, namely replay, masquerading and man-in-the-middle. This paper identifies critical weaknesses in the protocols proposed for V2I and V2V communication modes and proposes a new protocol suite as a countermeasure. Our protocol suite is composed of driver authentication, V2I and V2V key exchanges, information exchange, offline password change and vehicle complain protocols. Our V2I key exchange protocol facilitates handoff capability to ensure continuous authentication when vehicles move from the coverage of one roadside unit (RSU) to another. The protocol also assures detectability to denial-of-service (DoS). Our V2V key exchange protocol enables vehicles to verify the time-bounded validity of certificates of vehicles and integrity of keys. We use the random oracle model to prove the security of our key exchange protocols and prove various security attributes of the protocols informally. Tamarin tool is used to formally verify the security properties of our driver authentication and key exchange protocols. Performance comparisons show that our driver authentication and key exchange protocols assure lesser computation overhead and more functional attributes than the existing protocols. Simulation performance ensures the fast key dissemination capability of our protocol suite.