Let an old vet give you young'uns some advice.
1) Skills train even while you are offline, so make sure your skill queue is always full! At the beginning this will actually involve you logging in for 5 minutes every day, but eventually some skills take over a week. The skill queue allows you to plan ahead 24h without having to log back in to change skills. You guys have it lucky, this wasn't implemented when I started playing.
2) Make sure your clone is up to date. Always. Even in high-sec where its 'safe'. That is to say, always make sure that it can hold more skillpoints than you have. It is a relatively cheap expenditure, compared to your ship anyway, and there is nothing more frustrating than losing several days training time to some joker with more money than morals (like me).
3) The best money in Eve is literally scamming for it. And it's even legal. There are several things you CAN'T do, such as scam for real money, convince people to buy plex (in-game gametime) with real money for you as part of your scam or impersonate another player, but making promises then forgetting about them as soon as you have the cash in hand is perfectly legal. No matter how many times the victims threaten to 'petition CCP'. This isn't for everyone though, and don't feel bad if you can't bring yourself to cheat someone else out of their hard earned (space)cash.
4) On that same note, know that SCAMMING IS LEGAL IN EVE. If anyone asks you to give them money, accept that it is a scam of some kind. I don't mean trading on the market, I'm talking about any sort of deal that involves giving money where they can just walk away with it without giving you anything in return. This includes 'give me isk and I will give you double back' or 'give me isk and I will let you join this corporation'. In fact, if anyone EVER asks you for money in order to join a corporation, just accept that it is a scam.
5) Playing Eve solo SUCKS. If you are looking for fun, join a corporation. Yes, even as a fresh-faced newbie. Even if it is just a corp you cobble together with a few other newbies.
6) Get a buddy invite before you start. It extends your free trial time to 21 days and gives your buddy a whole free month. And it doesn't cost them a thing. Best way to make quick friends.
7) PvP is NOT some kind of evil creature. Don't be afraid to start pvp'ing early, even cheap ships can be very effective at their role. But NEVER undock anything (outside of 0.5 and higher sec systems) that you are not comfortable losing.
Get comfortable losing ships. It's going to happen, no matter your play style. Just remember that they are all just pixels. Learn from your loss, and next time do better.
9) Accounts never get deleted as far as I know. But gametime is cheaper if you buy in bulk, and really expensive if you only pay 1 month at a time.
10) Training stops if your accounts becomes deactivated (due to running out of gametime), but you don't lose anything. All partial training progress is saved, so you can change skills whenever you want.