I'll second that Mining and Missions are a side show to the main event in Eve - playing against other people. Faction War, Solo low sec, Null sec, Trading, whatever. Even the Null / Wormhole Relic sites are explored whilst under the threat from ganks from other players - that's what makes them fun.
As a PvE game Eve is considered pretty bad. As PvP though, absolutely amazing. I'd also second Pandemic Horde from what I've heard, if you want fleet fights and group play. You can be out in a fleet on day one. Anyone who tells you that there is a minimum skill point level to PvP is wrong. There might be a minimum to fly a certain ship a certain way, but I'm still flying T1 frigates, which you get handed out in the tutorials.
You might as well fly low skill point, poorly fit ships (cheap!) for your first PvP fights: your first fights will be lost as you get swamped by information and pressure, misclicking and piloting error, rather than any difference skill points, ships or modules make. The sooner you get through those quickly and cheaply, the quicker you can start to learn.
Don't play just before bed, I frequently find my self lying awake still buzzing from the last fight, having won a fight in low hull, flying out with my ship on fire, guns hot, prop mod burnt out.