In the second Gold shirt mission, the instructions tell you to sit at the helm... Then there are only a few buttons there, you click until you find out what does what.
Left to right, the 4 red helm buttons are:
Go to Warp
Go To Impulse
It's real easy, in that mission you just have to press the exit dock button, then watch the viewscreen as you leave orbit. Then a message will tell you to go to Warp Speed. When you go to Warp it will tell you to go to Impulse and then you will reach your destination and automatically enter the space dock.
You repeat that process 3 times to finish the Ensign Training mission, this takes you to all 4 planets.
To beam down to the planets, from the main menu you would pick the Sulu character, then load up the Explore All Decks mission, click on the AI Sulu to kick him out of the helm chair, then sit down and take the ship to wherever you want to go... There are coloured buttons at the top of the helm console, and each sets a course for one of the planets.
By default you will go in this order if you don't select a course:
But you can use a helm button to set a course to anywhere while you are in warp, and then you will emerge at that location.
Once in space dock, you go to deck 7, find a transporter room, click on the guy and you will beam down. Explore around, then click the transporter icon to beam back up.
In Explore All Decks, you can do this indefinitely, going to any planet and then beaming down, coming back up, going somewhere else, etc.
You can even take a shuttle from the hangar, and fly towards the planet, and you will eventually go through the surface and see the terrain and can land. When you take off, fly towards the North and you will emerge into space and see the Enterprise, where you can land the shuttle. Then go back to the bridge and take the Enterprise to another planet.
You can also run the helm from the deck 7 auxiliary bridge, and the deck 19 battle bridge.
Hope this helps, have fun!