Starting Out
Next, do the career agent tutorials! Seriously, do them! You can find their location by hitting F12, then clicking the Show Career Agents button (fourth button down) under the Support tab.
There are four career agents, one each for business, exploration, industry, and military. It's recommended that you do them all, since they all provide important information. If, for no other reason, do them for the money, ships and skill books they provide.
At this point you can start exploring different activities in more depth. New players typically begin by focusing on missions, mining or PVP.
Missions
If you want to continue down this path, you now have two choices. Your first choice is to find a regular agent that will provide you with more missions. To find an agent, use the Agent Finder button that's located above the list of agents when you're docked. A detailed mission guide can be found at http://wiki.eveonline.com/en/wiki/Missions_guide.
Your second option is to travel to the Arnon system and do the level 1 epic arc missions. The agent is located in the Arnon IX - Moon 3 - Sisters of EVE Bureau station.
Mining
Perhaps your best bet is to get yourself a Venture, stick some Miner Is in the high slots, Expanded Cargoholds and/or Mining Laser Upgrades in the lows, and whatever miners put in the mids.
Train industrial skills, save up isk, and move into bigger and better mining vessles with better fits.
PVP
If PVP is your thing, you have a few options.
Your first option is to find a corp that will teach you. There two resources readily available to find a corporation. The first is the official Eve Recruitment Center forums.The second is the in-game channel called Recruitment.
Another option is to join Red vs Blue (check out the in-game channel R-V-B for more information).
Your third option is to join faction warfare, which you can read more about it at http://wiki.eveonline.com/en/wiki/Factional_Warfare. Faction warfare was designed to be new-player friendly, but you may find that it's not necessary so.
Since it can take some time to find a good corp that will teach you PVP, your best bet may be to join Red vs Blue.
And More
There are a number of other things to do in Eve as well, such as manufacturing, trading or exploration. Check out this link for an idea of what activities are available.
Important Skills (some of the skill names have since changed)
Core Skills
- Electronics - Increases CPU
- Engineering - Increases powergrid
- Energy Management - Increases capacitor size
- Energy Systems Operation - Increases capacitor recharge speed
Defensive Skills
- Shield Management - Increases shield capacity
- Shield Operation - Increases shield recharge speed
- Hull Upgrades - Increases armor HP
- Mechanics - Increases hull HP
- Repair Systems - Increases armor repair speed when using an Armor Repairer
Gunnery Skills
- Gunnery - Increases rate of fire
- Rapid Firing - Increases rate of fire
- Surgical Strike - Increases damage
- Motion Prediction - Increases tracking speed
- Sharpshooter - Increases optimal range
- Trajectory Analysis - Increases accuracy falloff
Useful Mods
- CPU can be increased with a Co-Processor.
- Powergrid can be increased with either a Reactor Control Unit or a Power Diagnostics System (the latter also improves your shields).
- Capacitor can be increased with a Capacitor Booster (you must use cap booster charges with it), a Cap Recharger, or a Capacitor Battery.
Resources
http://www.isktheguide.com - Extensive guide on EVEhttp://wiki.eveonline.com/en/wiki/Main_Page - Official EVE wiki, including a 'Getting Started' guide.
http://lifeinneweden.blogspot.com/2011/02/eve-activities.html - Gives you an idea of activities you can do in EVE.
http://wiki.eveuniversity.org/Main_Page - The wiki of EVE University, a player run training corp for new players.