What are programming languages written in?
Programming language compilers and runtimes are written in programming languages — not necessarily languages that are older or are different than the one they take as input. Some of the runtime code will drop into assembly to access certain hardware instructions or code sequences not easily obtained through the compiler.
Once bootstrapped, programming languages can self-host, so they are often written in the same language they compile. For example, C compilers are written in C or C++ and C#'s Roselyn compiler is written in C#.
When the Roselyn compiler adds a new language feature, they won't use it in the source code for the compiler until it is debugged and working (e.g. released). This akin to the bootstrapping exercise (limited to a new feature rather than the whole language).
But to be clear, there is the potential (and often realized) for the programming language to be written in the latest version of its input language.
So what came first, and what was that coded in?
Machine code came first, and the first assemblers were themselves very very simple (early assembly languages were very easy to parse and generate machine code for), they were written in machine code, until bootstrapped and self-hosted.