A few pointers on skills:

• You do not grind Skill Points. Skills train constantly, whether you are logged in or not.
• SP spent is never SP wasted. Someday you might need those two levels of industrial you trained as a newbro.
• Training more skills allows you to do more things. Training them to higher levels allows you to do them better.
• Skill training speed relies on your character attributes, and whether they match with the primary and secondary attributes of the skill you are training.
• There are a huge number of skills that fall into several general areas. What skills you will want to train depends solely on what you want to do in EVE.
• Once you've chosen a few things to focus on, use EveMon to create skill plans.
• You can check the skills needed to fly fleet doctrine ships.

The training system in EVE is a bit different from other games. Most traditional games like World of Warcraft require you to grind “XP” to “level up.” In EVE, your character's skill progression advances constantly, whether or not you are even logged in. On top of that, there are no “talent trees” or other crazy point-buy systems to unlock by leveling up. Any player can train into any ship/role/job they want, provided they wait through the training time.

Despite what you may have heard, this does not lend an unfair advantage to older players; there are only so many skills that will affect any particular ship. A player who has played for 2 years might be able to fly the fearsome, projectile-turret-oriented Maelstrom battleship near-perfectly, but they might be outclassed while flying the missile-oriented Caracal Cruiser by a 3-month-old newbie who has trained the relevant skills higher.

EVE has this super cool training queue system that allows you to queue up multiple skills to train consecutively. As of Phoebe there is no limit to how many skills can be placed in the skill queue, but one limitation you will run into is that in order to place a skill in the queue, you must have its prerequisite skills trained already. You cannot work around this by putting your prerequisites before it, you have to have the relevant skills completed before you can place the skill in the queue. For example, if I want to train Covert Ops I, I require Electronics Upgrade V as a prerequisite. EVE will not allow me to add Covert Ops I to the queue until I've completely trained Electronics Upgrade V.

Every skill has 5 levels, each taking longer to train than the last. For example, level 1 of a skill might take 250 Skillpoints (SP) and about 15 minutes to train (training at 1000 SP/Hour), but level 2 would require 1500 SP and about an hour and a half, and so on. (Generally) having a skill at Level 1 means you've trained it to beginner levels, while Level 5 means you have it completely trained. For most skills, level 3 or 4 is a good compromise between effectiveness and time spent training. Training level 5 in a skill can take days, weeks, or in some cases months to train, I recommend newer players stick to training to level 3 or 4 until they decide where to specialize (except in some cases).

Skill points are preserved if you suddenly decide to start training a different skill, the progress you made will be there when you come back to it.

Training rate for any particular skill is determined by its primary and secondary modifying attributes. For example, most Gunnery skills have Perception / Willpower. Your training time for any given skill is related to your attributes in those two modifiers, specifically by the equation:
Points per minute = (primary attribute + secondary attribute/2)
In order to train faster, you must increase your modifying attributes; there are currently two ways to do this.

Implants are items that you plug into your character's clone in order to increase certain attributes. Remember, implants stay in your current body - if you jump clone out your implants will stay behind in your old body, and if you get pod-killed you will lose the implants currently in your head.
Test Free typically gives out sets of +1 attribute implants, and it is not a bad idea to constantly have one of these sets plugged in.

When you start out, you have 1 standard remap and 2 bonus remaps available. A remap is what it sounds like, it allows you to re-allocate all of your base attribute points. You should use your standard remap within the first few months of playing EVE (ask in deedeereddit for useful remaps for what you would like to train). After using your standard remap, you will not get a new one for 365 days. In addition you can use your 2 bonus remaps to change your attribute points anytime and completely independent to your standard remap. Using a bonus remap, DOES NOT change the counter of your standard remap. It is imperative that you do not waste your remaps. Use EVEMon and get help from someone in deedeereddit when choosing how and when to remap.

Skills are divided into 23 subcategories where within those subcategories lies the individual skills. I am not going to list and describe every skill in Eve Online, you can read the descriptions in-game. I will, however, briefly explain each subcategory and what they contain.