The update hasn't broken anything. The upgrade has. You must always run update before upgrade and treat upgrade with extreme caution. (At the very least, use the -S swtch to simulate first, better yet, follow the instructions at the very top of the original /etc/apt/sources.list)

It also looks like your /etc/apt/sources.list is broken or out of date which won't have helped.

What error messages are you getting when you try and run

apt-get install (package)

?