Where in the GCC source code does it compile to the different assembly languages?

The .md (machine description) files of GCC source contain stuff to generate assembly. GCC contains several specialized C/C++ code generators (and some of them translates the .md files into code emitting assembly).

GCC is a very complex program. The documentation of GCC MELT (an obsolete project) contains several interesting links and slides, notably refering to the Indian GCC Resource Center

Most of the optimizations in GCC happens in the middle-end (which is mostly independent of source language or target system), notably with many passes working on the Gimple representations.

The GCC repo is an SVN repository.

See also this answer, notably the pictures inside it.

The actual source code for GCC is most accessible from here:

https://gcc.gnu.org/svn.html

The software is accessible via SVN (subversion), a source code control system. This would be installed on many versions of Linux/UNIX, but if not on your platform, you can install the svn kit and then fetch the source using the following command:

svn checkout svn://gcc.gnu.org/svn/gcc/trunk SomeLocalDir

GCC is complex and would take significant experience to understand the nature of how the application actually compiles to different architectures.

In a nutshell, GCC has three major components - front-end, middle and back-end processing. The front-end processor has the component of the language parsing to understand the syntax of languages (like C, C++, Objective-C, etc). The front-end deconstructs the code to a portable construct which is then passed to the back-end for compilation to the target environment.

The middle part performs code analysis and optimisation, attempting to prioritise the code to generate the best possible output at the end of the full process. Technically, optimisation can occur at any part of the process as patterns are discovered during analysis.

The back-end processor compiles the code to a tree-style output format (not actually final executable code). Based on what the expected output is designed to be, the "pseudo-code" is optimised for using registers, bit-sizes, endian-ness, and so on. The final code is then generated during the assembly phase, which converts the back-end code into machine executable instructions.

It's important to note that the compiler has many options to deal with output formats so you can create output to many classes of architecture, usually out of the box. For cross-compiling and target compiler options, try checking out this link:

https://gcc.gnu.org/install/configure.html