Looks to me like you need a continuity tester to figure out which wire go where {each pair of wires goes to one speaker, I imagine each tied pair of pairs go to one room with 2 speakers , meaning a pos and neg for each speaker, to fin which go where, tie 2 together one at a time and then check them in the closet with an ohm meter}...
Anyway, You have a couple options, 1 you will need a source which can be anything, ipod, apple, cd whatever... next an amplifier
4z-75 2-Channel, 4-Zone Amplifier | Sherbourn Technologies not sure what your budget is... Or you can go with a d box and 2 ch amp
For only $15.88 each when QTY 50+ purchased - 4-Channel Speaker Selector and for an amp yuo can use
Dayton Audio APA150 150W Power Amplifier 300-812 or what ever 4 ohm stable 2 ch amp you want, upa200, ect.. {you want medium wattage under 200 and 4 ohm stable}....
Then for speakers you can do what ever you want, just make sure they are 8 ohm and mildly sensitive. Choose between built in or bookshelfs...
If I were doing it as cheep as possible, I would get a dayton apa150 amp for $160, a monoprice secector for 25, then buy a couple of these for under $200
For only $79.90 each when QTY 50+ purchased - 5.1 Channel Home Theater Satellite Speakers & Subwoofer - Black, use them inexpensive speakers for all the rooms maybe even wire in the subs {or sell them, or use them in a lil pc system, what ever}... That would cost around $400 all done, and although it wont be super hi fi or anything, it will get sound in all them rooms, amazon has ton of cheap little hangers that will work fine too...
In my house I use an emotiva 200 amp, monoprice selector and yamaha in ceiling speakers, running off zone 2 off of my HT preamp.... Never any problems, just turn it on and music everywhere...