After installing a program using a package manager where is the information about where the program was installed and in which menu category it can be found.
For where the program was installed (including all of the supporting files, so it can be a long list that you would need to search through)
dpkg-query -L firefox
For the menu item containing the package, look for a file in /usr/share/menu in the output of dpkg-query:
dpkg-query -L firefox | grep /usr/share/menu
In that file you should find "section=". The string after that text should indicate where in the menu system you would find the link to the application.
For information about what categories Ubuntu put the package into, you might try:
aptitude show firefox
which [application name] should give you what you're looking for.
If by "program" you meant the binary, it usually goes to /usr/bin
Resources go to /usr/share/
User specific configs usually are created in . in the user's home directory