How to self-learn data science?
Welcome to the site, Martin! That's a pretty broad question, so you're probably going to get a variety of answers. Here's my take.


Data Science is an interdisciplinary field generally thought to combine classical statistics, machine learning, and computer science (again, this depends on who you ask, but other might include business intelligence here, and possible information visualization or knowledge discovery as well; for example, the wikipedia article on data science). A good data scientist is also skilled at picking up on the domain-specific characteristics of the domain in which they working, as well. For example, a data scientist working on analytics for hospital records is much more effective if they have a background in Biomedical Informatics.
There are many options here, depending on the type of analytics you're interested in. Andrew Ng's coursera course is the first resource mentioned by most, and rightly so. If you're interested in machine learning, that's a great starting place. If you want an in-depth exploration of the mathematics involved, Tibshirani's The Elements of Statistical Learning is excellent, but fairly advanced text. There are many online courses available on coursera in addition to Ng's, but you should select them with a mind for the type of analytics you want to focus on, and/or the domain in which you plan on working.
Kaggle. Start with kaggle, if you want to dive in on some real-world analytics problems. Depending on your level of expertise, it might be good to start of simpler, though. Project Euler is a great resource for one-off practice problems that I still use as warm-up work.
Again, this probably depends on the domain you wish to work in. However, I know Coursera offers a data science certificate, if you complete a series of data science-related courses. This is probably a good place to start.


Good luck! If you have any other specific questions, feel free to ask me in the comments, and I'll do my best to help!