I have N three-dimensional vectors
(x,y,z)
I want a simple yet effective approach for clustering these vectors (I do not know a priori the number of clusters, nor can I guess a valid number). I am not familiar with classical machine learning so any advice would be helpful.
The general Sklearn clustering page does a decent job of providing useful background on clustering methods and provides a nice overview of what the differences are between methods. Importantly for your case the table in section 2.3.1 lists the parameters of each method.
The differences in methods tend to be based on how the knowledge of the dataset you have matches the assumptions of each model. Some expect you to know the number the number of clusters (such as K-Means) while others will attempt to determine the number of clusters based on other input parameters (like DBSCAN).
While focusing on methods which attempt to find the number of clusters seems like it might be preferable, it is also possible to use a method which expects the number of clusters and simply test many different reasonable clusters to determine which one is optimal. One such example with K-Means is this.
The easiest algorithms for clustering can be K-Means (if your three features are numerical) and K-Medoids (allow any type of features).
This algorithms are quite easy to understanding. In few words, by calculating some distance measure between each observation of the dataset, they try to assign each observation to the cluster closer (in distance) to them. The main issue with these algorithms is that you have to specify how many clusters (K) you want, but there are techniques such as the Elbow method or the Silhouette that allows us to determine numerically which value of K would be a reasonable amount of clusters.