Algorithm Suggestions for a Self Project
I would start with EDA, as usual. Plot stuff. For categorical data, what counts do you have? For quantitative data, perform the usual summaries and correlations. Since you don't have a question in mind already, I would then move on to unsupervised learning algorithms to suggest patterns in the data. Clustering is definitely a good option here (probably preceded by feature scaling). The unsupervised learning algorithms can lead into supervised algorithms, with all the associated statistical tests.

Without knowing your data, you simply cannot choose algorithms.