The first thing to learn about keeping yourself safe in EVE is knowing what areas of space are relatively safer than others. I believe I have heard that the new tutorials cover a bit of this, but I have yet to run through them, myself (too many bad memories of the old tutorial system). All the same, the labeling of systems is rather intuitive. Each system has a bit of text in the upper-left corner that includes a line for security status. The number on that line, between 0.0 and 1.0, tells you how much police and NPC naval force is present in the system. Systems with security ranging from 0.5 to 1.0 all have CONCORD ships present. These ships will respond to illicit combat anywhere in the system, including asteroid belts. The thing to remember is that CONCORD is like the real-world police in that they are reactive. If you get shot and have your ship blown up, CONCORD may not make it in time to save you, but they will ensure that the guilty parties get their ships destroyed as well.
The result of the guaranteed retribution of CONCORD (and it is guaranteed, as successfully evading them is considered an exploit), is that any would-be ganker has to carefully decide whether the potential reward of the gear you're carrying is worth the loss of the ships necessary to blow you up before CONCORD responds. The best thing you can do while hauling things around in empire is to ensure that your ship is not too tasty a target. If you are in a regular hauler, fit shield extenders, as this will buy you time, and when the gankers scan you, it provides somewhat of a deterrent, since it will take longer to burn through your ship. If you must carry more expensive cargo, then letting the ship afk autopilot is not a good idea. The longer you spend moving from gate to gate, the longer someone has to set up a trap for you.
The other most common tactic to prey on carebears in high-security space is called can flipping. Most miners in EVE will jettison their ore into a cargo container while mining, since the container has a capacity of 27,000 cubic meters. This makes it a lot easier to mine for awhile, then make a few trips with an industrial ship to bring everything back to the station. What the ganker will do here is take one piece of ore from the miner's can, jettison it into his or her own can, then transfer the rest of the ore to the new can. This action flags the ganker, so the miner and his/her corpmates can attack the ganker without CONCORD interference. Often, however, the ganker will be in a cloaked ship, so the miner may not realize what has happened. If they, in turn, take ore from the new can made by the ganker, they become flagged to the ganker, and can be freely destroyed. The trick is, when taking ore, always check the can. If you have cans displayed on overview, cans not belonging to a corpmate should show up as yellow diamonds, whereas your own cans should show as white diamonds.
There are other methods to target vulnerable players in highsec, but the above two are the most common.
Next, you have low-security space (0.4 to 0.1). In some ways, this is the most dangerous section of space, especially in terms of risk to reward. Lowsec has no CONCORD or naval presence. Your only NPC protection comes from sentry guns mounted around all the stargates and stations in system. These guns will fire based on the same principles as CONCORD, but they do far less damage, and a skilled pilot can easily fit a ship to withstand the fire from the sentries. As a result, you may occasionally see gate camps in lowsec, where a number of pirate vessels will sit around a gate destroying anyone who wanders by and simply tanking the guns. These are somewhat uncommon except in high-traffic areas, however. What is more common are the variety of pirates who hunt their prey in asteroid belts. With no NPC assistance available, a PvP fit ship can usually easily dispatch an unescorted mining or ratting ship.
The best means of protection in lowsec is an organized corp with enough members to provide a deterrent to pirates. Nearly all of my early EVE career was spent in lowsec as an industrialist, then later as a PvPer. This was possible because I was part of a large, organized corp that went out of its way to eliminate pirates in their "home" constellation. A pirate would ocasionally wander by, but would usually get sent home in his pod, if he was lucky. Another critical factor is monitoring the Local chat. Keep it in a separate window from your other chat channels, and check new arrivals compulsively. As soon as anyone shady comes into system, warp out. Don't wait to see if they're coming for you.
If you spend enough time in lowsec, you will probably eventually be destroyed at least once. The trick is to mitigate your loss through insurance, as was mentioned in the first Rogue Signal. Also, just to reiterate, never...EVER fly a ship you cannot afford to lose. EVE is a deadly place, and you will lose it, sooner or later.
Hopefully, this has helped some of the newer players, or prospective players, of EVE realize that while the bitter old vets might love to talk about how dangerous the game is, and how noobs are more meat for the grinder, a smart player can avoid some of the more common methods employed by player pirates.