Analyzing mobile usage. What kind of approach should I apply?
There are two straight forward (vanilla) ways without going for any fancy featurization: 

Clustering:

Run a clustering algorithm. Something like k-means should work well with this kind of a dataset. While doing this, I would not feed the day_of_week information into the clustering algorithm. 

I would suggest running k-means (after normalizing each of the columns). Choose a small number of clusters that is easy to investigate (or you could use the number of clusters that maximizes the BIC). 

Investigate the clusters to understand membership by day_of_week in each of these clusters. 

Multi-class Classification:

Treat the day_of_week as the response that you would like to predict. Build a decision tree of a fixed depth to predict the day_of_week given the columns. By examining this tree, you can easily tell, which decisions led to a set of leaves being labeled Sunday vs the set of decisions that led to a set of leaves being labeled Monday. These decisions will also help you understand the similarities between different days.