I would say AUC is the best overall metric for classification but does not have to be the only metric, accuracy is useful too. For reference you can check this Quora regarding accuracy vs. AUC:
They both measure different things, so they are complementary.
Accuracy: Measures, for a given threshold, the percentage of points
correctly classified, regardless of which class they belong to.
AUC: Measures the likelihood that given two random points — one from
the positive and one from the negative class — the classifier will
rank the point from the positive class higher than the one from the
negative one (it measures the performance of the ranking really).
Log loss can also be a good candidate as an overall metric, why can be read here from FastAI:
Log Loss vs Accuracy
Accuracy is the count of predictions where your predicted value equals
the actual value. Accuracy is not always a good indicator because of
its yes or no nature.
Log Loss takes into account the uncertainty of
your prediction based on how much it varies from the actual label.
This gives us a more nuanced view into the performance of our model.
RMSE on the other hand is a regression metric and should not be used for classification.