To begin, click "Start Game." Then click "Start New"
This will prompt you to select a game mode. There's sandbox, science and career. For this guide, we'll be taking the latter.
If you'd like, you can choose your flag. It will be visible on both your spaceships and the flags themselves you can put down on planets.
Once you start, the game will throw you straight into this map with dozens of buildings scattered throughout.
The middle one, which can be considered the heart of KSP, is the Vehicle Assembly Building. Here is where all the building of rockets takes place.
Right under that we have the Research & Development building. Here is where all the science happens and It will be very important later on to unlock both new technologies and building parts through Research Points which is unlocked by conducting experiments during flight and completing missions.
Left to that we have the astronaut complex. You should really never need to even click on this building hopefully, as It is only to replace your dead "Kerbals", AKA, the astronauts in this game.
Above that we have the spaceship hangar. Here you can construct the more advanced stuff such as airplanes as when you build, you need to do it sideways. This requires precise aerodynamics and a lot of experience with the Kerbal Spcae Program as It's physics and materialistic properties are not 100% up to par in real life.
All the way to the right we have the tracking station. Any launched spacecraft/airplanes will need to be tracked down, communicated with and subsequently recovered if possible, from here.
All the way up top we have the launchpad. All buildings constructed in the vehicle assemby building will be placed here and launched subsequently.
Last but not least, we have the mission control center. This building will be particularly vital and funnily enough, It is hidden from the standard view by about 70%. To spin around and see it, hold down the right mouse button.
To begin with, you want to head into Mission Command..
And grab all the available missions.
You can see what they require you to do and what the reward & penality will be for succeeding/failing it. These with this symbol, however, are the "main" missions and should be completed for maximum amount of money and science points.