Since this is the spoiler forum, I'll give you heavy detail just to get you a good start:
The first few missions you have to take from Juni. Do her missions until she gives you the access codes to open all the jumpgates in Liberty. When she gives you a break, ignore her. You don't have to decline, just don't go where she is telling you to meet her.
Explore, explore, explore. Go to every single station in every single system. Do random missions, talk to everyone, make loads of cash until you have the Liberty heavy fighter fully loaded and at least 10,000 credits and you have explored every system AND found every hidden jump hole and every hidden base. You might have a minor issue later if you spend too much time getting money in Liberty: The campaign missions are triggered by your wealth. If you have too much wealth, you might have to just decline some missions so you can explore the rest of the galaxy later.
Don't bother trying to get everyone to like you. Very quickly after Juni's missions, the whole reputation list gets turned upside down. Just do missions for cash or weapons or ships.
Don't fall in love with the Liberty heavy fighter either, you'll need to ditch it quickly...
Me no gun