So here's what I'm guessing...
You have 3 rooms with 2 sets of speakers in each room.
Each room has a wall plate in it. This wall plate is at normal 'switch' height. So, 40 inches or so off the floor. Perfect location for a VOLUME CONTROL! But, there is no volume control installed at any location. Instead, there is ONE white cable with 4 conductors in it. Red, black, white, and green. There are also two other wires in that box with two conductors each. They appear to have some color to them. Legally, they are supposed to be in a jacket, but it doesn't look like that is the case. Anyway, those wires go out to the speakers in the room. Silver for positive (red/white) gold for negative (black/green). The left speaker is typically wired to white and green. The right speaker is typically wired to red and black in a normal 4 conductor audio cabling setup.
So, this is something you can at least verify in every room. You can pull off the wall plate and just check to ensure that one of the two conductor cables is tied into red and black, and that the other wire is tied into white and green. From the photo you provided, it does appear that things are wired this way. But, it is good to check in all the rooms.
In the basement, you have the other end of those wires coming out of the wall directly, this is called your head end. I'm not sure how much length, or how organized those wires are. Sometimes there are labels on them for where they go, other times there are not. If I had done this, I would have put in a double gang low voltage box, and put in binding posts instead of just pulling the raw wires out of the wall. Then I would have labeled the binding posts for where they go and which is 'left' and white is 'right'.
These are the wall plates I'm talking about, and I'd get one with 8 binding posts (8 wires = 4 speakers total), and one with 4 binding posts...
But, those wires, are the distribution point for your audio. You have a TON of ways to now send audio from that head end location out to the various rooms. Since the rooms don't have volume controls in them, you need to figure out what and how you would want to listen to audio in those rooms. You also need to realize that a single stereo amplifier isn't designed to handle 3 speakers per channel. So, that needs to be addressed up front.
The simplest way is to get a stereo amplifier, and a speaker selector. Since the rooms have no volume controls, you can get a speaker selector/volume control unit. This will allow you to connect any source you would like (perhaps a Bluetooth receiver, or a Sonos Port, or a Wiim (mini or pro)). You can even plug your phone directly into the amplifier to get audio playback if you would like.
If you would like more control over each room, then there are options for that which will cost a bit more. Maybe a lot more. Depends on what you want to listen to.
But, at a basic level, you need an amp like this....
We Provide FAST and EASY Resolution!
www.ebay.com
A speaker selector/volume control like this...
And a source like this....
You would need some speaker wire to go from the amp to the speaker selector, and some RCA cables to go from the source to the amplifier as well.
All in, you can make this happen for about $300, for single source audio playback to the three different sets of speakers.
If you just want all three pairs of speakers to play back audio at the same volume, you should still use a speaker selector with impedance matching to ensure that it doesn't damage the amplifier. Something like this if you want to be as inexpensive as possible...
Likewise, you could skip the source and just plug your phone in or an old CD player. If you go cheap, you can get the cost down to just about $100.