Eve can be a harsh place, especially in the first few days. The game is unlike other MMOs and has a very steep initial learning curve (usually compared to a cliff inclined towards you). Nothing works the way you've come to expect from playing other games as you'll find out the moment you leave the starting station and try to use your usual keyboard movement keys to fly your ship. Which brings us to tip number 1...
1. Eve doesn't use keyboard movement keys. You'll have to learn how to use the UI, particularly the overview and right click menus to control your ship. Actually you can use traditional movement keys inside stations but you'll probably never do that once you become comfortable with the UI.
2. Right click is your best friend. When in doubt about how to do something, try right clicking on the object in question or even in empty space. Right clicking brings up a context menu which usually has the command to do exactly what you want.
3. Do all of the tutorial missions and career training missions. Besides teaching you how to do many different things, they'll provide you with ISK (game currency), loyalty points, reputation, skillbooks, fitting modules and even ships. Read mission descriptions carefully and do exactly what they tell you to do. F12 brings up the help menu, you can access the tutorials and career agents from there.
4. Regarding tutorial/training missions and steep learning curves, the part that gives new players the most trouble are the scanning missions. To save yourself a whole lot of time and frustration go watch this Probe Scanning Tutorial when you get to the scanning missions.
5. There's a Rookie Help channel. When you get stuck on something don't be afraid to use it, the people in that channel are there specifically to answer new player questions and can be very helpful.
6. Don't forget to keep your skill training queue full. You don't want to waste training time by having the queue finish and not be training anything for a while. Try to make sure the queue is full when you log off for the day and put a skill with a long training time at the end of the queue whenever you'll be offline for more than a day.
7. Security Status. Every system in Eve has an associated security status.
High Security systems range from 0.5 to 1.0 and while no place in Eve is completely safe hi sec is relatively safe. Players cannot attack you in hi sec for no reason without consequences (getting their own ships blown up by Concord). However if you attack them first or take their loot, you become fair game and Concord won't get involved.
Low Security systems range from 0.1 to 0.4, Null Security systems are 0.0 and lower (negative security status). Players can and will attack you in either without provocation or consequences. While there are significant differences between low and null sec new players can simply consider both dangerous places to avoid at first.
On the other hand besides a few specific systems most of low and null isn't as bad as the horror stories claim. Don't let the horror stories scare you from ever entering low/null, just be smart about it and whatever you do, do not go afk or fly on autopilot while in low/null.
8. Don't fly anything you can't afford to lose. Especially not in low/null. Don't carry anything too expensive in cargo either, not even in hi and especially not into a major market hub while on autopilot.
9. Can flipping. Cargo Containers (cans), ship wrecks and so on that belong to you are white. They're your's to loot, nobody else can loot them without giving you the right to attack them if you wish. Cans belonging to other players are yellow. Taking loot from a yellow can or wreck means you have aggressed the owner and he can now attack you without consequences. Blue cans belong to no-one and may be looted by anyone without aggression.
10. Don't forget to train Capacitor, Navigation and Fitting skills. They're important. You really can't go wrong training them all (except jump skills) to III early in your skills training plan. Or better yet, just train all the skills required for the basic Core Competency Certificate.
11. Fitting skills allow you to fit more and better modules on your ships. Engineering and Electronics are particularly important fitting skills and for combat pilots Weapon Upgrades is the next most important fitting skill. IMO if a character is going to be doing much flying around at all, the very first skills you should train to IV are Electronics and Engineering. They should be among the first skills trained to V too. Weapon Upgrades should probably be trained to III soon and IV eventually but you shouldn't need V for a long time yet.
Edit: Engineering and Electronics skills have been renamed CPU Management and Power Grid Management
12. Scams. Scamming (the in game variety, not the real life / real money variety), is allowed and scammers are everywhere. Scammers like to work with contracts and chat channels, especially local chat channels at major market hubs. The simplest way to avoid most scams is a) consider everything in chat to be a scam in the first place and b) read any contract very carefully before accepting it, especially if it's a contract you found in a chat link.