For multiple output neurons, to calculate the training error, in each epoch/iteration, you take each output value, get the difference to the target value for that neuron. Square it, do the same for the other output neurons, and then get the mean. So eg with two output neurons,
MSE = (|op1 - targ1|^2 + |op2 - targ2|^2 ) / 2
The training, validation and test errors are calculated the same way. The difference is when they are run and how they are used.
The full validation set is usually checked on every training epoch. Maybe to speed computation, you could run it every 5.
The result of the validation test/check is not used to update the weights, only to decide when to exit training. Its used to decide if the network has generalized on the data, and not overfitted.
Check the pseudocode in the first answer in this question
whats is the difference between train, validation and test set, in neural networks?