I tried to expand to places where no one was, which succedded for a bit, but i did eventually find new civs, but many of them had my allies between me and them so they didn't attack as much as my earlier allies.
But then they'll start attacking your allies and your allies will call for help and if you don't answer the call... yeah. Remember all the things that can degrade your relationships...
Not giving money when asked
Not answering your allies' call for help to fight a bio-disaster
Not answering your allies' call for help against pirates
Not answering your allies' call for help against Grox
Not answering your allies' call for help against an alien nation
Not answering your ally A's call for help against an attack from your ally B
Not answering your ally B's call for help against an attack from your ally A
Answering ally B's call for help against an attack from ally A
Answering ally A's call for help against an attack from ally B
Defending your territory from an attack from an alien race
...
The last point annoys me. I was trying to mend relationships with an alien race, but how am I supposed to do that when I have a permanent -100 diplo penalty called "You destroyed our ships!" ... Oh did I? Well you were attacking me I was defending my colony. You should only get such a penalty when you were the aggressor!
In fact, the same problem occurs in all other stages. I was trying to be "green" (friendly) in tribe stage after having been carnivorous (red) and adaptable (blue) in the first two stages... So I had a slope to climb on the graph thingy. I had to befriend all the tribes. But when they attacked me and I defended my village, my path on the evolutionary graph would take a steep dive towards the "aggressive" section. I managed to achieve my goal but it was a rather close call. I shouldn't get "aggressive" points when I'm defending!