What are recommended methods for multi-task prediction?
For the project, do you need to explain exactly how the input variables gave you the predictions? If so, trees based methods (R library 'tree', 'rpart') and Logistic Regression would be easy to explain to your stakeholder / lecturer.

If this explanation is not that important, or you could get by just showing the relative predictor importance, and you'd like to focus on accuracy. XGBoost is my default recommendation for multi-class predictions that I use at work on a daily basis. If XGBoost has too many parameters for your liking, you could try randomForest which only has 2 main parameters to tune. Both XGBoost and randomForest have functions for variable importance measures. (R Library XGBoost and randomForest)