Your best bet will be to look up the Reddit page of your power, and find the Discord link. The Reddits are generally bulletin boards, as the communities largely exist on Discord. They'll know the particulars of your power, as a few of them are a little different and the terminology can be obtuse. If you're interested in working for the power's benefit, you'll need to coordinate with others, as its easy to waste your effort or even cause trouble. If you're just after ranking up and the modules, then that part is easy.
Powerplay bonds are called merits, and the control system in the centre of a control sphere is where most of the action happens. You can see these in the galaxy map on the fourth tab, Powerplay button. The info card that comes up when selecting a system will show either controlled, or exploited, the latter being the systems around a control system. The main types of Powerplay activities are preparation, expansion, fortification and undermining. The game itself is about the collective management of Command Capital, or CC.
Undermining is going into an enemy control system, and killing ships with generic Powerplay names, like Imperial Supply. They differ for each power. They can be found most easily in supercruise, or res sites. Each control system costs CC to maintain. Undermining increases that cost, potentially leading to 'Turmoil' and the loss of control systems. You get 30 merits for each kill. You then take those merits to one of your own control systems, and submit through the station contacts page.
Fortification is increasing a control system's defence against undermining. You do this by going to your powers HQ, not just any control system, and picking up 'forts', a type of commodity, from the contacts station menu. The name differs for each power, and you can see this when you select a system in your right panel, Galactic Powers, Control tab. I'm ALD, so I'd pick up Lavigny Garrison Supplies from Kamadhenu, then take them to any system whose fortification progress was less than 100%. Anything over 100% is no benefit to the power, though you still get the merits for your effort. You get to pick up a certain amount every hour, depending on rank, though you can pay credits to speed it up. Each thing you deliver is 1 merit.
Preparation is what a power does in the cycle before an expansion. Same process as fortifying, but with a different commodity obtained from the same menu, and you can do it from the any control system, rather than only the HQ. The power that puts the most 'preps' into a system gets to attempt an expansion next cycle.
Expansion is different for different powers. Some powers are trade focused, like Felicia Winters, others are combat, like Hudson. If Winters wants to expand, she has to take Liberal Federal Aid commodities to the expansion system, similarly to fortification and preparation. For combat powers, they expand by fighting in special conflict zones with different names. For ALD its called a Crime Sweep, for Hudson it's a Military Strike, etc. These conflict zones award 10 merits per kill, and have to be taken to a friendly control system and submit through the station contacts menu. You're most likely to find enemy players here, as they can oppose the expansion.
That's just the tip of the iceberg, Powerplay gets pretty deep, and it can be a lot of fun. I strongly recommend getting in touch with the community, if you're interested in more than the rank bonuses. It's one of the few ways in Elite you can truly be part of something bigger. Edit: I just remembered squadrons exist now
