It really depends on what you want to do, as certain PVE/PVP skills aren't terribly useful to an industry pilot, and many industry skills are useless to a PVE/PVP pilot.
That being said, career choices can be broken down as follows:
Industry - Mining, Blueprint Research, Invention, Manufacturing (T1/T2/T3), and more
Combat - PVE (Missions/Incursions/Sites) PVP (Ganks, Pirating, Factional Warfare), and more
Trading - Trade in station and make spacebucks or haul goods around
There are various niches in each of these career "paths", but each has an overarching idea to the types of skills you would want to train if you were to head down them.
Combat is easily the most daunting, because there's so much to do and so much to train. In my opinion, the best thing to do is focus on one race at least until you have a handle on how that race works, then it's relatively easy to branch out and train other race's ships/guns.
That means if you're Amarr, train for Amarr Frigate, Amarr Cruiser, Amarr Battleship etc. (It should be noted that as of January 7, 2013, The Destroyers and Battlecruiser skills are not split up by race, but the makers of Eve have stated that they will be split in the near future). Find out which weapon type your race uses, and train the corresponding skills. Amarr use lasers, Minmatar use Artillery and Autocannons, Caldari use missiles and hybrids(rails/blasters), Gallente use hybrids and drones.
More importantly however, is if you intend on being a combat pilot, train your capacitor skills and your tanking skills. Every race needs to use capacitor for something, and training those skills will never be a waste. The most important Cap skills are Energy Systems Operation and Energy Management. Tanking depends mostly on the race of ships you're flying. Amarr and Gallente (mostly) armor tank, whereas Caldari and Minmatar (mostly) shield tank.
The armor tanking skills Mechanic, Hull Upgrades, and Repair Systems will serve you well.
Shield tanking is slightly more skill intensive, with the main skills being Shield Operation, Shield Management, Shield Compensation, Shield Upgrades
Another set of insanely useful skills to train includes Engineering, Electronics, Weapon Upgrades/Advanced Weapon Upgrades. These skills will allow your ship to have more powergrid, more cpu, and make your weapons use less of cpu and powergrid. It is impossible for ships to be fit optimally without training these skills, so you will never regret them.
An important avenue that is useful for almost every single pilot (especially gallente), is drones. There are drones that do almost everything in the game, and if you intend to be a combat pilot, you will want them. Skills to use drones include Drones, Drone Interfacing, Scout Drone Operation, Mining Drone Operation, and many, many more.
As for industry, there are less globally important skills to have, as each has it's niche. If you intend to mine, train mining, astrogeology, mining barge to get started. If you intend on refining the minerals you mine, then learn Refining and Refinery Efficiency. Most mining skills are found in the industry tab or science tab.
If you intend to manufacture goods, Production Efficiency is vitally important.
Invention is a more complex career that involves various skills ranging from science and industry to electronics/engineering.
Trading is the easiest career path to explain since you pretty much train all the skills in the trade tab, and do market research and buy stuff to sell later. The recommended skills are Broker Relations, Marketing, Trade, Daytrading and Procurement.
I feel it necessary to put a disclaimer here and say that I'm no expert in any of the careers I mentioned above, nor am I an expert in Eve. Nor is my answer a comprehensive guide of all the skills required to do any of the above mentioned activities at peak efficiency. This was only an introductory look at some of the skills that will always be useful for certain avenues in this game. I also glossed over less complex (skillwise) careers like trading and invention.
