Greetings fellow citizen, I too serve the royal princess! I don't think that's the same thing as belonging to a faction, however. I am still a little confused regarding PowerPlay in general.
Playing on the PC, I had a nice base system and station, that I favoured, Federation, of course, and spent a bit of time allying myself to the Federation linked factions in the neighbourhood. I dabbled with PowerPlay, but never really got deep enough into it to see results. From what I recall, it's a good way to entrench yourself into a political system, with various 'rewards' dependant on you doing certain varied tasks. It seemed to me to help with the roleplaying I was doing, and it certainly increased the number of NPC encounters I was getting, as I was now a target for the other political groups.
My recommendation would be that you start off doing some missions for factions aligned to the major faction (Fed, Imp, All) of your choice, to get some reputation up in that allegiance (it does help getting similarly aligned missions with better rewards when you do so).
Sort out whether you want to base yourself in a particular area of the bubble, or are happy going anywhere, and then decide if you want to get into PowerPlay.
Just my opinion, but it worked for me.