Vehicular ad hoc networks (VANETs) play a vital role in the success of self-driving and semi self-driving vehicles, where they improve safety and comfort. Such vehicles depend heavily on external communication with the surrounding environment via data control and Cooperative Awareness Messages (CAMs) exchanges. VANETs are potentially exposed to a number of attacks, such as grey hole, black hole, wormhole and rushing attacks. This work presents an intelligent Intrusion Detection System (IDS) that relies on anomaly detection to protect the external communication system from grey hole and rushing attacks. These attacks aim to disrupt the transmission between vehicles and roadside units. The IDS uses features obtained from a trace file generated in a network simulator and consists of a feed-forward neural network and a support vector machine. Additionally, the paper studies the use of a novel systematic response, employed to protect the vehicle when it encounters malicious behaviour. Our simulations of the proposed detection system show that the proposed schemes possess outstanding detection rates with a reduction in false alarms. This safe mode response system has been evaluated using four performance metrics, namely, received packets, packet delivery ratio, dropped packets and the average end to end delay, under both normal and abnormal conditions.