The game begins with your character being dumped onto a random planet at the edge of the galaxy. And with nothing but a busted starship for company, you are told to seek out materials to repair two key pieces of said ship -- namely the launch thrusters and the pulse engine. Now before you set out to do anything else, check the controls in the menu and mess around with the inventory to get a feel for how things work. Because No Man's Sky is as much a survival game as it is an exploration game. The survival element is more pronounced in the beginning however due to a very limited inventory space.
Plus keep in mind that besides your jetpack (and your character's stamina after running I suppose), nearly every piece of your equipment has a finite amount of juice, and must be recharged from time to time. So it is imperative that you stay well-stocked up on all the necessary resources to stay alive. Fortunately most of these resources are found in abundance in most planets, so you won't really die unless you ignore the warnings to recharge your life support systems.
Also, if you were to check the round distress beacon thing somewhere close to where your ship is, you will be given a choice between accepting guidance from the Atlas or not. That said, you still need to fix your ship either way. Remember to take a look at your multitool as well and fix both the scanner and analysis visor.
Now for the starship, you need four Carite Sheets to repair the launch thrusters. It requires 50 units of Iron to craft one Carite Sheet. The pulse engine on the other hand is a more complicated fix, needing two Carite Sheets, 200 units of Heridium and 20 units of Zinc. Press down on the left stick (L3) to use your scanner and locate mining spots for these elements, until you have what you need. The third objective is to refuel and requires some Thanium 9. Find these and you should be good to go.
Again make sure you're stocked on Thanium9 as each take off costs 25% of your total fuel so you will need to recharge often. Destroying debris during space flight gives you plenty of these so it's not hard to find. In any case, feel free to stay on your first planet and look around. The "?" that occasionally appear when you use the scanner are points of interest that are worth checking out. Fly your ship over to the area if it's too far away to reach on foot. You can scan animals with the analysis visor by pressing L2 as well. And finally in regards to saving your game, this is automatically done when you enter then leave your space ship, and also at certain beacons that are indicated with a "?".