Classifying transactions as malicious
This problem is popularly called the "Credit Card Fraud Detection"

There are several classification algorithms, which aim to tackle this problem. 

With the knowledge of the dataset you possess, the Decision Trees algorithm can be employed for detecting malicious transactions from the non-malicious ones. This paper is a nice resource to learn and develop the intuition about fraud detection and the usage of basic classification algorithms like the Decision Trees and the SVMs for solving the problem.

There are several other papers which solve this problems employing algorithms like Neural Networks, Logistic Regression, Genetic Algorithms, etc. However, the paper which uses the decision trees algorithm is a nice place to start learning.

what would be a good place to host the 600 or so GB of data for cheaps?

Aws S3 would be a nice, cheap way to do that. It also integrates nicely with Redshift, in case you want to do complex analytics on the data.