So, you may like mining or you may like PvP but right now you just want to make a little money? That's no problem. Running missions is an old hobby in EvE and one that can serve your wallet well. You should have already run some tutorial missions so I won’t bore you with the basics of how to do that. If you haven’t noticed it already the agents that give you missions have two numbers associated with them The first number is the level which takes into account your standing with the Faction and the actual Agent to decide if you are allowed to use them. The second number is the quality of the agent which determines the reward amount (it isn’t a direct correlation so don’t panic if you like a -10 agent).
As a new character your frigate may struggle with some missions but should get the job done if you set it up well. When it comes to setting up your ships for missions one of the best places to look is the EvE forums. Here you can advance from the work of others as well as ask your own questions about specific setups.
After you have your ship ready you will want to find an agent. One of the easiest ways to do this is online at Eve-Agents. You can search based on various criteria to help you find an agent that is right for you. If you have already downloaded EvEHQ there is an agent finder built right in which you may find handy as well.
Okay, so you have your ship and you have accepted a mission from your agent, now what? Well, you're going to run that mission of course. Before you do though you may want to take a trip over to EvE-Survival to check out the mission reports. Here you will find recon information from other pilots who have flown the missions you may be tasked with. You can use this information to better prepare your ship or to find out exactly which enemies you need to attack.
Once you have had some time to advance your skills and have a few missions under your belt there is quite a lot to out there. If you feel like venturing into null-sec chaising the big bounties on those NPC battleships then you may want read about Chaining Rats. On the other hand if you need a little bit of passive income and would like to put those hours of missioning to use again you will be very interested in using R&D Agents. You may want to keep this in mind when looking for your first few agents: not all factions have good R&D Agents so if you would like to take advantage of this you may want to start with a faction that does.