Some words of advice:
1) Don't touch yellow wrecks or containers. They belong to someone else and you can and most likely will be shot by the owner
2) Stay within 0.5 to 1.0 security space in the beginning. 0.4 is already the PvP zone and you will get shot
3) Do not fly ships you can not afford to loose.
4) Mining sucks
5) Get a corp. Eve is much easier and more fun when played as a team. Corp members also help you with getting new ships, advice and with the basic orientation in Eve
6) Do not fly ships you can not afford to loose.
7) Be aware: Scamming, ganking, killing etc. is 100% legal in EVE. Basically everything that works within the game mechanics is legal. If you buy something per contract, double check if it's really the thing you want to buy. CCP's mantra: Any harm to you, which could have been prevented by using your brain for 2 seconds, is your own fault
8) All races can do everything and fly everything. Get a feeling which way of playing you favor: PvE missile spam and active shields? Caldari. MadMax feeling with duct tape, solar sails and rusty ships for PvP pwnage? Minmatar (your way should be Rifter -> Thrasher -> Rupture -> Hurricane)
9) Do not haul (expensive) stuff around while flying afk in your untanked industrial. You'll get suicide ganked, looted and laughed at, not necessarily in that order.
10) Do not fly ships you can not afford to loose.
Never forget: EVE is a cold, harsh place. Welcome and enjoy it
PS: Sounds worse than it actually is, EVE is great. If you have questions or need tips, shoot me a PM.