Read the tutorials & do the 3 career mission chains.
Hit F12 and look at the bottom where it says "show career agents." While you may not identify with all 3 career categories, each will give you free stuff and a better understanding of the game.
Eve is a "sandbox." There is no linear progress down a preset path.
It has a steep learning curve and can thus be frustrating for new players who don't know all the choices and opportunities ahead of them. Embrace the steep learning curve because the depth and sandbox aspect of Eve is what makes it so great.
Basic initial options:
- the epic mission arc
- the militia
- build standing by doing missions with a Gallente corp
- mine, manufacture and trade (i.e. be a spreadsheet junkie)
The "epic arc" is a set of new missions designed for new players. It's a very significant chain of missions which take you around the entire empire regions dominated by the 4 major factions (Amarr, Caldari, Gallente and Minmatar). You can make good money as a new player, as you work your way through these missions and learn more basics. It starts with Sister Alitura who is located Arnon IX - Moon 3 - Sisters of Eve (station). Check your journal.
The "militia" is an opportunity to engage in faction warfare with an emphasis on PvP missions. Be sure to read about joining factional warfare and that you understand the full implications. This puts you at open war with Amarr and Caldari factions. There are special missions, events and locations. WARNING #1: this will be extremely frustrating for new players and cost you a lot of money that you don't have yet, because PvP means losing ships. WARNING #2: this will remove you from your starter corp (i.e. CAS).
Doing missions with Gallente NPC corporations will open up higher level missions and new opportunities. When you do missions with a Gallente-friendly NPC corp, you will gain standing with that corporation as well as with the Gallente faction. High standing with NPC corporations will cut down on station charges and open up new opportunities like creating Jump Clones. Completed missions will also get you Loyalty Points (LP) which can be used at LP Stores in corp stations.
- Pick a corporation based on the kinds of missions you want to do. If you like combat missions, you may want to investigate Federation Navy or the Federal Intelligence Office. You can browse the Gallente corporations using People & Places. "Fed Navy" is popular because it has a lot of agents and has many stations with Cloning Facilities (which you can use with 8.0 personal standing to create Jump Clones).
- Find an agent by looking at the "agents" tab in the Corp Info (i.e. Federation Navy), or by looking at the map ingame (set filter to show agents), or by using a nice website like eve-agents.com which also lets you sort by "required standing."
- You'll start with lower quality Level 1 agents. Levels 1-5 indicate difficulty. The "quality" of the agent represents the scale of rewards for that level (i.e. L1 Q18 would be an agent with high rewards for Level 1). As your standing increases, you'll gain access to higher quality and higher level agents.
- When picking an agent, consider the location carefully. Look ahead. Don't go into "low sec" (i.e. 0.4 security or below). Consider a location where there are several agents available, or soon to be available as you gain standing.
- When doing combat missions, you may want to decline "faction missions" against Caldari and Amarr because that will lower your standing with them.
- Frigates and destroyers are good for Level 1. Cruisers are good for Level 2. Battlecruisers for Level 3. Battleships for Level 4. Salvaging combat missions is great money.
- You can read about your combat missions at eve-survival.com where they will also tell you the types of damage the mission involves. This is very useful. We tend to "armor tank" Gallente ships, which means we mostly ignore our shields and focus on making our armor stronger by increasing their resistances to specific damage types while also using armor repair modules.
For more info on basic mining, see Mining 101.
Passive skill training may be counter-intuitive for gamers at first.
Get Evemon (the most useful eve tool of them all).
It will help you manage training plans and is also useful for looking up items, ships and skills. Combine this with EFT, which is another useful tool that let's you play with ship setups. You want to particularly look at how Evemon will show the benefits of Learning Skills as well as examining the "battleclinic" links at the top of ship information pages.
Every skill that you learn has a primary and secodary attribute that they connect to. Some skills harness your intelligence and wisdom attributes. Others may harness your perception and willpower attributes.
The higher your attributes then the faster you can train related skills. For each of your attributes, there are corresponding Learning Skills to increase those attributes. Get it? Perception of 30 will speed up training for new ships as compared to a character with only Perception of 10.
This is why veteran players tend to focus entirely on Learning Skills when they make new characters, because it will dramatically speed up training on other skills. THIS ISN'T PRACTICAL FOR NEW PLAYERS because they need to train more fun things while they try out the game.
Once you've committed to Eve and have played for a week, you need to look ahead with a tool like Evemon to figure out what your priorities are. It would be wise to get the basic Learning skill along with the other basic learning skills up to Rank 3. Rank3 for those skills is fairly quick and worth the time. Once you've picked up more fun abilities, you can start slipping in more learning skills. Once you hit Rank4 in a tier1 of learning, you can quickly pick up 3 more ranks in the next tear (i.e. Spatial Awareness 4 --> Clarity 3).
Use Evemon. Use EFT. Ask questions. Fly safe.