I am no expert, but I wanted sector space travel in my first mission also and I found out that you cant really travel in sector space more than once during your mission. When you accept the mission, you can travel to the initial system, but that's it. All other maps are instant.
I think that's a limitation in the system at this time. If I'm wrong, someone please correct me.
This has been talked about before. (search it) I will explain briefly...
There are work arounds, if your first task has multiple objectives. each objective can take place in a different system. Its not perfect and you can't give the standard 'exit' system prompt. This means you have to me EXTREMELY clear on how the mission proceeds.
I am also by no means an expert, however I've done some playing around and watched plenty of tutorial videos.
If the first transition is to a social map like Earth Spacedock, then you can have another system door transition after that. Plus if you have a 'complete all' objective before the first map transition then you can have multiple system door transitions in parallel (like a sector patrol mission).
To my knowledge, what you cant do is have multiple system door transitions one after the other (after the first social map), which is a shame.