Did your friends purchase the game or did they pirate it (v1.05)? Basically, you all need to have the same version of the game in order to play it together. Everyone should download and install Impulse and then register the game with it in order to patch up to the latest version, which is v1.191 for regular Sins. Also, note that you can see the version you have installed by opening the game and looking at the lower right of the screen. If some of your friends haven't purchased Sins, note that it's on sale for $4 as a download over Impulse right now, but that sale will probably end in a few hours.
Trinity is just a fancy name for the compilation pack, or a fancy way of saying "Sins of a Solar Empire Complete (with both of the expansions)". In your main Sins folder you will find three different .exe's. You might want to create shortcuts for them or a special Start Menu folder for them:
- Sins of a Solar Empire.exe
- Sins of a Solar Empire Entrenchment.exe
- Sins of a Solar Empire Diplomacy.exe
If your friends only have standard Sins then you guys can only play standard Sins together.