So, the first thing to do is an inventory of your rooms and list them out. Which room it is, and how many speakers are in each room. I have a huge system with 20+ zones in it, so I get things like this:
Family room - surround sound with video, 3.1 audio.
Kitchen - stereo, audio only
Morning Room - stereo, audio only
Deck - stereo, audio only
Garage - stereo, audio only
Playroom - stereo, audio & video
So, go through and make a list like that. Sounds like most rooms are stereo, but your family room may have a set of rear ceiling 'surround' speakers, and then you can use your Yamaha to provide 5.1 audio with you providing the front left/right/center speakers and a subwoofer. But, hard to know without that inventory list.
Then, you go through the process of identifying speakers. This can be done with a 9v battery, but if you are the DiY type, then maybe hooking up your Yamaha to a stereo pair (one pair at a time) and finding each speaker in the setup.
Speaker wiring should be coming out of the wall in pairs, or groups of 4 wires. If they are NOT labeled, you will want to label them so this never happens again. Red/black is one pair, white/green would be the other pair. Typically red/black is the right speaker in the room, white/green is typically the left speaker in a room. Red and white are positive, black and green are negative.
More photos of the wiring, the wall, and what that bundle in your hand is going to would be helpful. Don't need photos of the Yamaha, we can look up the RX-V677 online.
The RX-V677 is a 2-zone receiver which can run your main surround A/V zone, and send audio to a secondary zone. You will need some additional equipment, but you aren't there yet in actually needing that equipment, so first, let's just figure out what you have, then we will figure out what you need.
Yes, a installer can come by and figure it all out at typical professional labor rates, and it will take them a few hours to get it all figured out and properly labeled, but if you have time, then this is work you can do yourself.
Once things are organized coming out of the wall, one nice thing you can do is actually take it all back inside the wall and use wall plates and label them as to where every single connection goes to. That way, should you leave, the next owner gets a clear picture of what is connected in the home.