I'm sure you're downloading away right now (or not, idk). Make sure you pick up the helmet on the pedestal before entering the ship and after you're in the ship you can use the tab key to free the mouse and select options. Should be fairly straight forward from there.
It will take another eight hours or so. What happens if I don't use a helmet?
Nothing. By that I mean literally nothing. You'll get in the ship and nothing will happen, you'll just sit there. The lore is that the ship hooks up to the helmet and the simulation is just a VR for your character. I'm sure you'll ask "why don't I just spawn with the helmet on?" and that is because the helmet is shoehorned in after the original hangar module released. Also, some people want to sit in the ship and admire the cockpit without the menu in their face.
The ship the game gives you will be an Aurora and I won't lie, Auroras suck balls. They'll be a lot of other Auroras since you and the other freebies will be playing, but I'm sure a Hornet will eventually come by and push your shit in for giggles. For a good comparison imagine playing E:D in an arena mode where you're in a Sidewinder and other players have Vipers. Auroras have no port or starboard shields. Shields in SC are not bubbles, there is port, starboard, aft, and forward shields. Ignore the shield power section of the HUD, even if it shows full power to forward shields, it is just lying to you, nothing happens.
Also, keep in mind the three flight computer options (similar to E:D's flight assist). You'll have "Coupled", "G-Safe" and "Comstab" enabled by default. "Coupled" means the engine will direct thrust in the direction the nose is pointing (disabling it results in the engine attempting to keep your heading regardless of nose direction). "Comstab" automatically steadies the ship, useful to keep on, but sometimes you'll want it off, especially if some of your thrusters are damaged. "G-safe" limits your ship's rotational speed and forward thrust to human-safe levels. If you turn it off it is possible to red out and then black out from g-forces, but you can also maneuver more quickly.