It's all keyboard-driven. Pressing UP accelerates your ship FORWARDS. Pressing LEFT and RIGHT turns your ship respectively. Note that you DO have inertia... you'll drift until you hit the thrusters again. The exception is while you're in hyperspace - it takes fuel to travel in there, so you'll glide to a stop if you're not continuously thrusting.
Selecting "Navigate" puts you back into maneuvering mode. When you chose "Starmap" and picked a destination, you told the autopilot to go to that spot in hyperspace. Don't do that just yet - the first part of the game takes place entirely in Sol.
That warped black/red screen you were wondering about? That's your ship crossing from normal space (the black) into hyperspace (the red).
You only start with ONE lander and ONE Earthling Cruiser in your little fleet - see the icons in the window on the right? The diagonal white thing is the Earthling Cruiser, and the little triangle under the C in "Vindicator" is your lone lander. The big gray thing in the middle is your flagship, and if you look closely, you can see what modules you have installed on it (currently very few). Fuel is also used for landing on planets - more massive planets take more fuel. A ship's health is measured by its remaining crew - lose the last one, and the ship or lander explodes.
Selecting "Manifest", IIRC, lets you see how much cargo you have, both in resources and in items, as well as allowing you to reassign crew. You start with one fully-crewed Earthling Cruiser (18), and one crew-pod's worth on your flagship (50, the number listed at the bottom of the status window).
During the lander sequence, you press the primary fire key (Space, I think; whatever you were using to select stuff with) to fire your lander's stunner, and the secondary fire key (Shift or Control, I really should dig out my copy) to have the lander take off again. The green dots in the status window are your lander's crew, the red bar is how much cargo it has, and the blue bar is how much biological data. Don't hesitate to pull the lander out if it's taking a beating - you can always land again with a fresh crew (at least, until you run out of crew). There's three hazards - wildfires which burn in a curvy line, earthquakes which show as expanding white circles, and lightning which shows as, well, lightning. The harsher the environment, the more of each you see.
