Any mission you do for the ruling faction inside a Federation system counts towards your reputation and will lead to you eventually getting your system permit to Sol.
Find out which of the local factions has the most influence. Then check out the bulletin board for missions issued by that faction. Don't agree to do illegal missions, as they may damage your reputation. Don't even do them for the ruling faction.
The safest missions (the ones that are actually possible to finish without getting desperate) are as follows:
"Fetch" missions: "Bring x amount of cargo y back here". Only accept them if you already know where to get cargo y. Thrudd's Trading tool can help you (some consider this cheating, but just because the internet doesn't exist in the game for some unknown reason, it doesn't mean that it shouldn't.)
"Haul" missions: "Transport the cargo we provided you with to starport x in system y". Easy-peasy. Just check whether you can actually reach the designated star system with the amount of cargo they want to provide.
"Courier" missions: "Deliver message/pamphlets/important plans to starport x in system y". See "Haul" Missions.
"Kill" missions: "Kill x amount of pirates in system y or z" Go to the designated system and either hunt wanted ships at the NAV beacon, in an extraction site inside a planetary ring (if available) or hop from USS to USS hoping to run into pirates. When you have killed enough of them, "Objective completed" will appear on your HUD and you can head back for your reward.
Don't bother with the "kill x traders", "kill x smugglers" or "kill x bounty hunters" missions, as there is currently no way in game to determine whether one is a smuggler or bounty hunter. Don't accept "kill authority ships" missions, as they get you wanted. Yes, even if you do them for the government. Murder is murder, no matter your contractor. The Senate will not protect you for doing their dirty work.
There is one variation that is a legitimate "kill" mission: "Destroy Warzone. Go to conflict zone in system x and destroy y amount of military ships." These missions are bugged, though... sometimes they will send you to a system that has no conflict zones. Check beforehand whether there is actually a conflict in the designated system. Also: Be careful which side you pick upon entering the conflict zone.
