Making cash from exploring:
Map your Basic Discovery Scanner to a firing group ("4" to bring up the system panel on the right, "n" to cycle to the next firing group). Once you got the D. Scanner selected, jump to a system that is partially or totally unexplored and hold down the trigger until you hear an awesome pinging gong sound. This will reveal all of the objects within 500 lightseconds.
More distant objects can be found with an upgraded scanner, or by looking for the bright points of light that move against the fixed background stars and flying towards them. Once you get within 500ls, use your scanner again.
You can also do more detailed scans by targeting an object marked as "unknown" and keeping it in front of your ship until the "Scanning..." finishes and the name of the object displays.
You can sell the data at any station that is at least 20 lightyears from the surveyed system. You get more cash for systems with more stuff, and the most cash for detailed scans of valuable planets, like earthlike or mineral-rich ones.
Finding places to explore:
There are lots of unsurveyed systems all over the place. Some are even inhabited but don't have their data added to the cartographic database. As near as I can tell, what other players are doing doesn't have a huge impact on how much cash you make, but I haven't been that far outside human space so that might change if you're exploring things 1000s of lightyears away.
Surviving Exploring:
I do a lot of exploring along the way on my long-distance trading runs. It's a great source of supplemental income. If you're specifically just exploring, you're going to want a fuel scoop, so you don't risk being stranded. A heatsink launcher is also useful, eventually, in case you accidentally end up too close to a star (some of the larger ones can be dangerous). Eventually, you'll want an auto-repair unit for when you're really far away from inhabited space, but you probably want to wait until you have a bigger ship than a Sidewinder for that.
Also, having a longer jump range helps.