As other answers have mentioned, you can indeed overload operators (by definining special methods in the class you’re writing, i.e., methods whose names start and end with two underscores). All the details are here.
To complete the answers to you questions: you cannot define new operators; but << is not a new operator, it’s an existing one, and it’s overloaded by defining in the class the method __lshift__.
As a historical note, this is also pretty much the situation in C++ — but the exact set of operators you can overload differs between the two languages. For example, in C++, you cannot overload attribute access, .; in Python, you can, with __getattr__ (or __getattribute__, with different semantics) and __setattr__. Vice versa, in Python = (plain assignment) is not an operator, so you cannot overload that, while in C++ it is an operator and you can overload it.
<< is an operator, and can be overloaded, in both languages — that’s how << and >>, while not losing their initial connotation of left and right shifts, also became I/O formatting operators in C++ (not in Python!-).