I have a couple buddies who are a week or two behind me in EVE Online. I figured I'd type out a couple tips here so they're in one convenient place. Most is wisdom gleaned from elsewhere, but I've thrown my own thoughts in there, too.
Almost all of this can be done on a trial account. And if an MMO hasn't hooked you by the end of the free trial, it doesn't deserve you, your time, or your money.
Take Time on Your Portrait and Name
You'll be staring at it as long you're playing EVE--and there's no way to change it. So get something you want to look at. One other thing to consider: Much of the time, you'll be looking at a small, icon-size version of your portrait. Make sure the three main elements (your head, your shoulders, and the background) aren't equally light or equally dark. Shoot for light/medium/dark, split up however you like.
Likewise, you're stuck with your name. So make sure it's cool.
(Note from the future: As I was typing this, Tycho from Penny Arcade was thinking in parallel. Check it, yo.)
Play Through the Tutorial
The "Crash Course" tutorial is pretty good. It takes a couple hours, depending on how much you explore the interface. When you're done, you should have some cash, a ship--and most importantly, you'll have a decent idea of whether EVE is for you. If you're intrigued, keep going. But if that wasn't interesting, move on to some other game.
Play Through the "Career Advancement" Missions
You'll get three dudes who each have an eight-part quest line. Play all three, all the way through. The order doesn't matter, probably--although I did the soldier one first and got a combat-worthy frigate that I used in the mining and trading mission strings. You'll get several ships and some new skill books with these missions. And more importantly, you'll get a taste of the different roles you can play in EVE.
Buy Skill Books
At this point, the gravy train of free stuff is largely over. Visit the market and buy whatever skill books you want/need. The certificates screen is a useful guide. Certificates don't do anything in and of themselves. It's a useful way to describe "skills you'll need if you want to fly that badass cruiser," though.
Learn to Learn
Your first priority should be to acquire skills that enable game activities. For example, if you want to outfit that new Tristan with a scout drone, go ahead and get the Drones skill and spend the minutes required to get level 1 in it. But early in the game, you don't need higher levels of the skills. The NPC ships you face early on...well, they aren't exactly ninjas.
Instead, train the Learning skill and the five Attribute-specific training skills (Iron Will, Eidetic Memory, and so on) up as high as you can, as soon as you can. Why? They reduce your training time for every subsequent skill you learn...including themselves!
When you visit the market, you'll also see five other Attribute-specific learning skills that are crazy expensive (from your vantage point as a noob, anyway). Those stack with the basic five learning skills, and you'll max those out when those prices aren't quite so daunting.
Don't Fly Anything You Can't Lose
Yes, get insurance, and yes, you should watch your accumulated skill points so you know when to upgrade your clone. But don't rely on clones and insurance payouts to get you back on your feet. Your hull may be insured, but you'll lose all those fittings and whatever's in your hold if somebody blows you up. And if they destroy your escape pod, your clone will wake up just fine, but you won't have any cyberware you bought. ('Cause it's a clone, see?)
My initial thought is to keep three-quarters of my wealth in my wallet and fly around with the remaining quarter. But that's just a rule of thumb to guide my newbie explorations. Once I'm established, a better metric would be something like, "Fly around in something that takes no more than x hours of foo to replace." I'm not sure what my tolerance for x is. And "foo" is an activity: mining, courier missions, Traveller-style trading, missions, "ratting" (beating up NPC pirates and taking their stuff), or whatever my chosen "pleasant and reasonably lucrative grind activity" is. Speaking of which...
Find a "Pleasant and Lucrative" Grind Activity
This task is where I am in the game right now. I'm spending an evening or two doing each of the following, keeping track of how much ISK I make. It's not the most scientific undertaking, and the "answer" doesn't have to hold for more than the medium term. It's as much an exploration of different roles as it is an exercise in finding economic efficiency.
* Mining, probably in a 0.6 or 0.5 system. I have neither the firepower nor the friends to go for the really lucrative stuff. But I'm a noob, so that's just fine.
* Missions. Telling my starmap to show me systems with available agents was a revelation. Lots to do there.
* Ratting. Gotta spend a little on salvage sensors, etc., but this might be right up my alley as a grinding activity.
* Trading. Again, I have neither the capacity nor the knowledge to make a killing on the market. But me and my freighter can probably make a "bruising."
* Contracts. The courier stuff is what I'm thinking here. You can't do this stuff on a trial account. This is a "can you read the contract and adequately assess the risk?" as much as anything else.
Out of Context: "Why do I have to be the salt shaker?"
Music: Jeff Buckley, Grace