So since PSN is back up I figured I would try out the multiplayer part of Brink however when I select PSN it just lets me choose to play with my friends or see if I have any notifications...How do I get into the lobby to select other servers to play on?

First off there is no lobby

you go in to freeplay and choose your settings (try competitive so bots dont appear) and let people join or you will join an existing game. Same with challenges and campaings ( which i dont think it has a competitive option)

well the campaign, you can play versus or co-op.
all you need to do is change the how many players which is currently set to solo.

Playstation Network tab is only there to manage your friends.
you can only play online via Campaign, Challenges and Freemode. just make sure you are not playing 'Solo'