When new to a verticial industry, I like to go with the Naive Bayes algorithm to learn about the data. It is the least complex algorithm, but ALWAYS test with the other appropriate algorithms as well. Not all algorithms return the same results.
The tasks you have stated do not really intuitively sound appropriate for the Time Series algorithm but it is tough to answer the question without seeing the data model.
Decision Trees is a nice swiss army knife.
Association rules is good for "People that bought X also bought Y".
Clustering is commonly used for customer segmentation and classification but can be used in other ways too.
Sequence Clustering is common used to predict the next event in a sequence of events (i.e. web site clicks).
Hope it helps.