As soon as you are allowed to work you can become self-employed. Therefore you need a Gewerbeschein (trade license) for Kleingewerbe (small trade). You can get this in your town for 20 Euros (you only have to pay for it once). With this paper you can do small services (e.g. gardening) as they are supported by Aiden for example. Other professions might be more complicated than that when it comes to the question which papers you need. If you need specific information for certain professions you can just ask and we can try to answer your question.
The general advantage of being self-employed is that you do not need to find an employer. Instead you can start working on your own right away.