I've never edited admin files before, but I need to edit the table used to list my posts in the admin area (on the All Posts page). Can someone please point me to the right file to edit? I've looked in the wp-admin directory but I haven't found a file with this table in it, so far.
You do not need to and absolutely should not be editing a core WordPress file to achieve this. There are almost certainly hooks for whatever it is you are looking to accomplish. You can take a look at questions regarding custom columns (such as Add column to pages table) or, if you'll excuse my linking to something from myself, I have some slides regarding that same topic: http://slides.helenhousandi.com/wcphx2012.html#13
I'd strongly advise against editing WordPress core files.
For one, it isn't unlikely that a secondary bug will occur when part of the core is changed without proper knowledge of its structure and dependencies. Also, your changes will get overwritten with every automatic update you run, so you'd have to backup the edited file(s) and/or snippet(s) and re-edit the core after every WP update.
If you need a customized display of all posts, I'd rather recommend writing a simple plugin (or incorporating it in your theme), that handles the desired table output.
See these codex pages for reference: