My drupal 7 messed up after installing a module and I just get an error, how do I debug it and trace the source of that error? If I have a module that I installed and its causing a conflict with another module, how do I trace it down to see which is causing the conflict? Let's say the error message doesnt' give any useful information in finding the culprit module. How would one go about find the module causing the issue?
You can use the debugging methods mentioned in the comments, especially ddebug_backtrace. 

Another, more manual option, is, to disable other enabled modules one by one, until the error goes away. Once you know the culprit, search the issue queue of that module (and the new module) and/or use the debug statement (or drupal_debug from the devel module) to narrow down the problem. 

Some pointers on how to debug:
https://drupal.stackexchange.com/questions/545/how-do-i-debug-drupal
How do I debug my Feeds import not importing a field

And then of course, you should look at the web-server logs (apache, etc). 
Another way to debug issues like that, when you get the white screen of death, is, to execute a drush command on the command line - like cc or something like that. Often, the error message gets printed to STDOUT.