I meant drop a ship core, yeah. You should be able to see it in your navigation tab/on the screen.
Building neat things probably shouldn't be an early goal. Your first goal should probably be to set up a permanent home base. You can do that on any planet or station by putting a faction module on it, creating a faction for yourself (under the options tab of the faction screen you can open by pressing 'h') and pressing r while looking at the faction block. Click on "enter faction signature", the radar signature should turn green and if selected as a target (with f) it will say your factions name after the name of the station/ship/planet. After that, enter the faction blocks menu again and select "make faction home" (it needs to be a station or planet; ships can't be your home base.)
Your faction home is indestructible under normal circumstances. Set up a room indoors where pirates or whatnot can't snipe you and you're good.
For early game I'd go about mining for a little while. If you press 'p' it will open the galaxy map, showing the sector you're in at the center. Use wasd+eq to navigate the selection around. Normally there would be at least one gold ring orbiting the star In the starting system, though it isn't guranteed; gold rings denote asteroid belts. You can mine asteroids, and sell what you got at a shop for quick cash.
Don't go on early game expeditions without a radar jammer, though, and have it activated at all times. Pirate ships will ignore you if you have a radar jammer on your ship, enabled, but turrets won't so watch out. Expect to die regularly. Remember that exiting a ship resets the radar jammer, so don't get out of the ship to use other objects if you're not absolutely sure its clear. Access them through build mode if they're on the ship.
Sorry, missed this; a good Intermediate goal would be to set up a defensive turret network around your home base. Since anything docked to a factions home base is indestructible, they don't need to be all that tough, just able to hit their targets and cause permanent damage. This might be completely unnecessary if pirates never attack, but then you can just pop the turrets off home base and put them on your assault craft when you're ready to go bust a station.