This paper develops an integrated framework comprising deep learning methods for the real-time prediction of transient instabilities, as well as their severities, that might evolve in a power system following a critical contingency. In the first stage of the framework, for an early prediction of transient instability following a fault, phasor measurement unit (PMU) post-fault measurements are fed into a convolutional neural network (CNN) classifier that was previously trained using a cost-sensitive instance weighting (CSIW) approach. By the proposed approach of weighting the loss function based on the severity of unstable instances for training the classifier, the sensitivity of the classifier to instabilities is enhanced, and thus, the classifier of the first stage is made more unlikely to miss the cases that could lead to a wide-spread blackout unless a corrective action is taken. Moreover, a second supportive tool, which is the severity prediction function, is proposed to be integrated with the transient stability prediction function to assist the decision-making process. If the power system is found to be unstable, the severity of the fault inducing instability is successfully predicted in the second stage by using a long short-term memory network (LSTM) to increase the awareness concerning the severity level of the disturbance on which emergency control actions could be planned. The results are compared with model-based methods as well as data-based methods, such as over-sampling and under-sampling. The superiority of the proposed method is demonstrated on the 127-bus WSCC test system and Turkish power system.