You can get fast money early on with smuggling, since it makes the most of the small cargo of the Sidewinder. Upgrade your FSD when you can, as you'll find some missions almost impossible due to the small jump range you start with. The Sidewinder doesn't really do anything that well, so you'll want to upgrade when you can. A good general purpose one that is pretty cheap is the Cobra, and it's so useful many people keep it as one of their main ships.

If you want to do exploring, you can get a decent budget explorer for just over 2 million, such as this Hauler. When outfitting an explorer, you want most things to be low class and D rated, as these will be the lightest and give you the biggest jump range. However, the FSD should be the best possible. It's also a good idea to get an A rated power plant to make fuel scooping without overheating easier. For things that have no weight, just get what you can afford. You'll spend a lot of time fuel scooping since there's no stations when you head away from civilisation, so get the best fuel scoop you can afford. An advanced discovery scanner is also recommended, or you'll be limited to 500ls/1000ls for discovering objects. A detailed surface scanner will improve profits.

For actual exploring, when you arrive make sure your throttle is at 0 in case you arrive at a neutron star or black hole, or you could crash. Fire your discovery scanner (you can set this to a fire group) to find objects, then open the system map to see what's there. If you see things you want to scan, target them then fly towards them. When in range, it will begin to scan automatically. Once done you can move on, and sell the data the next time you get back to a station. The detailed surface scanner works passively, and will gather extra data automatically as long as you have it and the module is powered. You can refuel at the following star types: O B A F G K M. If you filter the galaxy map by star type, these are the first 7 types listed.

Also, don't be too worried about getting destroyed. When destroyed, you'll automatically eject and jump to a nearby station, where you can claim on your insurance. Look on your right hand panel, and it should show how much money you have as well as the rebuy cost. As long as you have more money than the rebuy cost, the rebuy is all it will cost you to claim the insurance on your ship and get it all back (minus cargo). Always keep your credits above the rebuy cost. And remember that the starting Sidewinder and equipment (which will say that it is loaned) is free.