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The end of the career tutorials – which should take you no more than a day or two to complete, if that – marks the end of the truly necessary beginner missions. There are more missions to be done for various factions: working for the Sisters of Eve can be very profitable, or you can begin to earn loyalty points by doing missions for a certain faction. You are also free to begin mining, trading, producing or pirating, though it may be some time before you are effective at any of these professions. Some general tips for what happens next:
Avoid low-sec and null-sec space at first. These are systems that are marked 0.0-0.4. You can be attacked by others players in PvP here and they will not be forgiving merely because you are new. Wait until you have a stronger or faster (or more expendable) ship and until you are aware of the risks.
Join a corporation. There are many corporations (similar to guilds) that will be willing and even happy to help new players get started and may provide you with money, skills, ships and assistance. It is also through a corporation that you will do most of your long-term advancement in EVE.
That's all there is to starting out in EVE Online. Though it has a reputation for being difficult, most of this is because of the openess of the world - there is no set path to follow. Once you've got your bearings, however, there is a great deal to do! Hopefully this guide has helped you on your way to becoming the next great captain of industry or fleet commander. Good luck!