With so many paths to choose from and so many ways to enjoy Eve Online, one of the biggest challenges is deciding what skills to spend the time training. I've covered before and stick by that training the CORE skills is still a good idea, because the benefits they provide will help you with whatever path you choose. To figure out what the CORE skills are, open your player toon and then click on Certificates. The Core Certificates are integral to all paths (with maybe the exception of someone that only does business trading) so you can't choose wrong there.
Aside from Core skills, there are so many paths available to you to choose from as I'm sure you know by now. There is no way to get all the skills faster with the exception of adding implants to reduce your training time. I highly recommend you find or earn the ISK to buy basic implants because over time the time you save will add up. My one point of advice that I would like to ensure that you get out of this article if nothing else is to pick a goal and to try and stick to it for a bit. If you want to be a miner, focus on mining skills. If you want to earn ISK through planetary interaction, focus on PI skills. If you want to run Logistics in a fleet, stick to it. Picking to earn one mining skill here, another PI skill there, and maybe a remote repper skill here, will make you horrible at all three and not very good at anything. The same applies for what type of ship you should be flying.
Initially I trained up Frigates to level 3, then went to destroyers level 3, then cruisers level 3, then battle-cruisers level 3. Looking back on this decision I wish I would have gotten good at one thing first before I spread myself all over the place. Granted it's nice that I can fly a battle-cruiser when I need to and a Frigate when I need to, but I can only fly each of them decently, and not great. It doesn't take very long to train these skills up to level 3 (typically less than a day for each skill to level 3), but those are precious days in the beginning. Since then I have decided that I want to be a stealth bomber pilot. The reason I chose stealth bombers is because it's at the frigate level plus the skills that I gain to fly (and fit properly) the Tech 2 advanced frigates will carry over in the future to cruisers, battle-cruisers, etc.
Being able to fit a ship well is way more important than just flying the more expensive ships. Granted I could spend 3 days and be sitting in a battleship, but I wouldn't be able to properly fit and fly it for months, which would result in me losing lots of battleships, all of which cost much more than frigates. Losing a stealth bomber will cost about 30 million for the ship, and probably another 10-15 million for the fittings. Losing a battleship costs about 80-180 million for the ship, plus another 30-50 million for the proper fittings. I'd rather make a 40 million ISK mistake than a 200 million ISK mistake.
People are commonly in a rush to get into a battleship, but then find that they don't have the skills to put whatever they want onto it and the result is having to put lots of CPU, Power grid, or capacitor modules to be able to fit it or stay cap stable, which means they weren't able to fit the proper armor, weapons, or shield to help keep the ship alive when the DPS starts coming. If they had taken the time to get their core skills up, then they would have been able to have a little more CPU, PG, and Cap to play with, resulting in them being able to fit armor or shield modules on their ship. And once they fit the armor or shield mods, they'll notice they ALSO take more CPU, PG, and Capacitor and are back to square one at not being able to fit everything they want onto their ship. Flying a battleship with T1 drones and medium turrets is a joke, and you have no right even be in that ship. Step down, get back in the battle-cruiser, cruiser, destroyer or frigate and fit it properly. You'll stay alive longer and be more useful to your team.
One of the major reasons I like Eve Online is the team play aspect. With a good corp there are lots of organized joint operations that are fun to participate in. Even an advanced frigate pilot can find a very useful place in a corp, and be responsible for helping out in a big way on many missions. A 1 month character flying a battleship is as good as dead on a mission and only really offers themselves up as cannon fodder, useful but not as useful as flying an advanced frigate and flying it well. The time will come when you can move into cruisers, battle-cruisers, or battleships and by the time you get there you'll be able to bring a lot more to the table.
Don't get me wrong, it's not a super bad idea to spend 2-3 days being able to fly cruisers and battle-cruisers early on, just don't spend too much time on them until you have a solid foundation. I know a player that rushed to be able to fly a Tech 3 Proteus but didn't take the time to ensure that he can deal or receive lots of DPS. He definitely is able to offer a good assistance on missions but compared to a skilled Proteus pilot he is probably only able to provide half the help that he could be. Take the time to train skills to level 4 or 5 (5 is not always a good idea, it depends on the time to train versus what you get in return)
To sum up; be a great miner, or manufacturer, or pilot, or explorer, or logistics but don't try to be everything, because the result will be that you're not that good at anything and you will get frustrated with the game and quit.
Cheers, and fly safe.