First off, if you are brand new to EVE because of the Ascension update that allows you to play for free, let me say: WELCOME! Its a wonderfully rich sci-fi simulation that has few borders on what you can do with your time. You're not cast in a mold and forced to dance to the game's tune, you are free.
Secondly, let me give you some advice from a long time player (ten years) to help you get started.
DO THINGS WITH OTHERS
EVE is a simulation rather than a game and as such the experience, like real life, benefits greatly from being a shared experience. And not only does the experience fare better, people will help you and teach you things you don't know, provide assistance in whatever you are doing, and share a laugh when things go wrong (as they inevitably do as you learn).
So once you get in game, do the tutorials, get familiar with enough to move around and operate your ship.... find a group to join. There are many organizations, big and small, that will gladly welcome you into their ranks and help you progress from newbie to journeyman. Decide what you want to try out, find a corp or channel that does that (in your timezone), and introduce yourself! No matter how new you are, you are a valuable resource to these organizations because your growth helps their growth; your success helps everyone's success.
And if its not working out, don't be afraid to try another group. No one holds a grudge against someone looking to find the right fit for themselves (unless you depart with a bunch of corp assets!).
On that note, if you are looking to try PvP in a casual friendly atmosphere, Aideron Robotics is recruiting!