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Exploration
If you prefer to work alone, then exploration might be your favoured option. Fly solo through system after system, scanning for the abandoned structures that can be looted for valuable materials. It’s a great chance to really hone your solo skills; in low or nullsec, avoiding gate camps and explorer hunters will teach you the importance of the d-scan and how to avoid the bad guys, and the occasional big score will make up for any losses.
How To Do It
Read the Brave Dojo's Guide to Exploration, which contains detailed explanation of the skills you'll need, the types of ships and fits you should use, and how to perform the two key aspects of exploration: scanning and hacking. For another, more personal look at exploration in nullsec, you can read Brick Hunter's guide to exploration.
If you're starting off in High Sec, you will have less to worry about in terms of random encounters with other players, but correspondingly smaller rewards from the exploration sites.
Where To Do It
Everywhere! Exploration is probably the activity that will see you roam the furthest in EVE, as you search through system after system for elusive data or relic sites to loot. In a small, nimble ship outfitted with a cloak, you can easily dip into low-sec and nullsec with ease, increasing the possibility of finding a big score to bring home.
What To Do Next?
The materials you obtain from exploration are used in the manufacture of items like rigs, so will fetch a good price on the market (if you don't want to start manufacturing yourself). You can either sell them to one of the alliance's buyback services, or haul them to a market hub system.