How do I find out what all symbols are exported from a shared object?

Do you have a "shared object" (usually a shared library on AIX), a UNIX shared library, or a Windows DLL? These are all different things, and your question conflates them all :-(

• For an AIX shared object, use dump -Tv /path/to/foo.o.
• For an ELF shared library, use readelf -Ws --dyn-syms /path/to/libfoo.so, or (if you have GNU nm) nm -D /path/to/libfoo.so.
• For a non-ELF UNIX shared library, please state which UNIX you are interested in.
• For a Windows DLL, use dumpbin /EXPORTS foo.dll.

objdump is another good one on linux.

On *nix check nm. On windows use the program Dependency Walker

see man nm

GNU nm lists the symbols from object files objfile.... If no object files are listed as arguments, nm assumes the file a.out.

The cross-platform way (not only cross-platform itself, but also working, at the very least, with both *.so and *.dll) is using reverse-engineering framework radare2. E.g.:

$ rabin2 -s glew32.dll | head -n 5 
[Symbols]
vaddr=0x62afda8d paddr=0x0005ba8d ord=000 fwd=NONE sz=0 bind=GLOBAL type=FUNC name=glew32.dll___GLEW_3DFX_multisample
vaddr=0x62afda8e paddr=0x0005ba8e ord=001 fwd=NONE sz=0 bind=GLOBAL type=FUNC name=glew32.dll___GLEW_3DFX_tbuffer
vaddr=0x62afda8f paddr=0x0005ba8f ord=002 fwd=NONE sz=0 bind=GLOBAL type=FUNC name=glew32.dll___GLEW_3DFX_texture_compression_FXT1
vaddr=0x62afdab8 paddr=0x0005bab8 ord=003 fwd=NONE sz=0 bind=GLOBAL type=FUNC name=glew32.dll___GLEW_AMD_blend_minmax_factor

As a bonus, rabin2 recognizes C++ name mangling, for example (and also with .so file):

$ rabin2 -s /usr/lib/libabw-0.1.so.1.0.1 | head -n 5
[Symbols]
vaddr=0x00027590 paddr=0x00027590 ord=124 fwd=NONE sz=430 bind=GLOBAL type=FUNC name=libabw::AbiDocument::isFileFormatSupported
vaddr=0x0000a730 paddr=0x0000a730 ord=125 fwd=NONE sz=58 bind=UNKNOWN type=FUNC name=boost::exception::~exception
vaddr=0x00232680 paddr=0x00032680 ord=126 fwd=NONE sz=16 bind=UNKNOWN type=OBJECT name=typeinfoforboost::exception_detail::clone_base
vaddr=0x00027740 paddr=0x00027740 ord=127 fwd=NONE sz=235 bind=GLOBAL type=FUNC name=libabw::AbiDocument::parse

Works with object files too:

$ g++ test.cpp -c -o a.o
$ rabin2 -s a.o | head -n 5
Warning: Cannot initialize program headers
Warning: Cannot initialize dynamic strings
Warning: Cannot initialize dynamic section
[Symbols]
vaddr=0x08000149 paddr=0x00000149 ord=006 fwd=NONE sz=1 bind=LOCAL type=OBJECT name=std::piecewise_construct
vaddr=0x08000149 paddr=0x00000149 ord=007 fwd=NONE sz=1 bind=LOCAL type=OBJECT name=std::__ioinit
vaddr=0x080000eb paddr=0x000000eb ord=017 fwd=NONE sz=73 bind=LOCAL type=FUNC name=__static_initialization_and_destruction_0
vaddr=0x08000134 paddr=0x00000134 ord=018 fwd=NONE sz=21 bind=LOCAL type=FUNC name=_GLOBAL__sub_I__Z4funcP6Animal

You can use gnu objdump. objdump -p your.dll. Then pan to the .edata section contents and you'll find the exported functions under [Ordinal/Name Pointer] Table.

Usually, you would also have a header file that you include in your code to access the symbols.