The training missions push you to getting a few skills by handing out skillbooks, but there are many more skills. You'll see and read about ships, so the ship skills at least should be on your radar. The certificates system could, in theory, be useful for learning about the existence of skills. But as of 2013, certificates are ill-thought out, and basically a joke. And in any case they are not exactly obvious either. If you are lucky, you'll get into a corp where some experienced player can mentor you.
What about the solo guy? I was a solo guy, more or less, just me and my small group of friends. The way we discovered skills was via EVEMon. Then we'd tell each other.
EVEMon is a standalone program. It stands for the EVE character monitoring (and planning) program. Or something like that. It's freeware. Anyone can download it at evemon.battleclinic.com. It's useful for many things, but in particular you can use it to discover what skills exist in the game that might be something you should train now. This can save you a lot of time, as versus attempting to find out about skills in game or via the web.
To use EVEMon with your particular character, you must generate a so-called "API" for that character. You do that via the web, here. Generate a full API key. Start EVEMon, then select "File/Add API Key...". Fill in the information from the web page to this dialog. (Cut and paste are handy here.)
Now you've got your character in EVEMon, what can you do? Well, the main use is to plan skills. We'll get to that in a minute. But you can see all sorts of info about your character, and this is what EVEMon shows on each character's tab. Take a minute to look at the sort of info you can get via API.
OK, let's talk about skill plans. Select "Plans/New Plan", and input a name for your new plan. You should now see the planning interface to EVEMon. If the "Skill Browser" tab is not selected, select it. Look at the skill tree on the left side; it shows a list of skill categories. Each category can be opened to show a listing of skills.
|I see some skills to get|
Most important for the newbie are skills that you don't know, but you have all the prerequisites for. These are shown with a green plus sign. You can train these skills starting right now; all you need is the skillbook to inject. Most such skills are desirable, but you'll want to select the skill in the list to see what it does. In the example, I have selected Evasive Maneuvering. A synopsis of what it does is shown in the text box in the upper right. In this case: Improved skill at efficiently turning and accelerating a spaceship. 5% improved ship agility for all ships per skill level. 5% agility per level? That's huge! Why did nobody tell you about this? Well, I am telling you, or EVEMon is. Go get it.
Now, get some ISK, and fly to one of the "training" stations which sell skillbooks. Get EVEMon going, and start down the list. Select every green-plus skill and see if it looks useful. If you cannot figure out what a skill does from reading its synopsis, then click the button "Show what this skill enables...". This will pop up a window that displays other skills that the skill is a prerequisite for, and items that require a certain level of the skill to use. If you still can't determine what the skill does, then it's time to hit the EVE wiki and maybe the web to find out if it is useful. If the skill is useful, buy the skill book and train to level 1. If the skill is reasonably useful, plan to train it to at least level 3, and probably level 4.
Generally speaking, a combat pilot will want to get any skill he can in Drones, Electronics, Engineering, Mechanics, and Navigation, and Spaceship command. You can focus on missiles, in which case get skills in Missile Launcher Operation (but still look at Gunnery for Weapon Upgrades and Advanced Weapon Upgrades.) Or focus on Gunnery. You won't need skills in Corp Management, Industry, Leadership, Planet Management, Social, or Trade. At least not for combat; those are money-making skills. Science has some skills everyone will want.