You must have a starship. Once you have a starship, you enter it, enter the control room, and then 'Enter cockpit' (an option under the Control tab at the right side of the screen in the members view).
From there, there's a button labelled 'Travel' below the map. Press that. If you're in a city, then you can press 'city' under the map. This will allow you to move around the city, OR (and this is what you're looking to do), 'Ascend'. That takes you into the planet's atmosphere. Then, afte rabout an hour's travel time, you'll be in the atmosphere. From there, you can press 'Planet' or 'Orbit' (Can't remember which it is at this time). Once you've reached the system map, you can either fly around the system (using Travel, sublight) or go to another system (by using Travel, Hyperspace).