Basically, what everyone is saying, is true. The only objectives in the game, are your own. If you want to be the best industrial pilot in EVE, then by all means; likewise, if twisted metal and frozen corpses is more your speed, maybe PVP is for you.
Some people choose to do the Epic Arc missions, to follow some of the story line and history of EVE, and new Eden, and the rewards from said Arc lines, makes it very tempting to do so!
To answer your question about the agression countdown, there are essentially three types of agression countdowns: one for NPCs, meaning you attacked a faction (namely pirates) and for that countdown, any NPC of that faction, will engage you on sight.
The second, is the standard PVP agression countdown timer, where you will be denied access to dock at a station for approximately 60 seconds. And ofcourse, for the duration of that countdown, the person you engaged, or stole from, can engage you without CONCORD interference. The third Countdown, refers to Global Security Status, where you are considered a Criminal to all of new Eden, and will be engaged by police, and gate guns/station guns on sight. I hope that answers your question.