Do all factions oppose each other?
All factions oppose each other, however, that doesn't necessarily mean they are hostile towards one another. As I said earlier I got a 40 merit fine for killing another Federal powers ship. Take a look at each power, see what their ethos is and how they like to do business. For instance. Zachary Hudson is all about combat whereas Felicia Winters is more about using money to achieve her goals. Choose one that fits how you would like to play. As to where to start, look at an opposing factions control systems and decide which one you want to try undermine then go and attack supply ships, security ships and any other faction ships in that system. Once your done head back to any control system under your own factions control to claim your merits.
One last thing...your weekly earnings are calculated at 8am GMT every Thursday morning. Your rating will be determined by the total of what you earned during this week and any merit you had carried over from previous weeks...the higher your rating the more you will get out of Power play. Carried over merit is subject to decay which means you can't just rest on your laurels if you want to advance.
Take a look at my explanation
here in the Guides and Tutorials section.