"Waste of time" is subjective. You're family is more important than your sound system. It sounds like you have your priorities straight.
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For the record, speakers have the most influence on sound quality, this is where you don't want to skimp.
According to Pioneer your receiver puts out 80wpc into 8 ohms. The 140wpc rating is into 6 ohms.
The Sony are rated at 150 watts, 8 ohm. This is peak or max. You can cut max power handling in half and get an idea of RMS. In this case it's 75. Your receiver would run these just fine.
You are correct, the front three should ideally be of the same brand and series speaker to keep the sound stage uniform. The rear surrounds can be something else, although Bose would not be on my list. The Polk sub will not do much at 100 watts, you can do better for not much more money.
You mentioned buying a piece at a time to increase your overall budget. This would be a good option. You can get by without a center or surrounds if you get better speakers for the fronts.
To be clear on this, ideally, you should listen to speakers before you buy. Not every speaker sounds the same, it depends on your personal taste. That being said...here's one opinion.
Front sound stage:
http://www.ascendacoustics.com/pages/products/speakers/cmt340m/cmt340m.html
Sub:
http://www.amazon.com/BIC-America-F12-475-Watt-Subwoofer/dp/B0015A8Y5M/ref=sr_1_7?s=electronics&ie=UTF8&qid=1420196680&sr=1-7&keywords=subwoofer
Rears:
http://www.ascendacoustics.com/pages/products/speakers/cbm170/cbm170.html
The Ascends have a great sensitivity rating, 92db, which means even a small receiver will power them very well.
Total speaker package price, $1100 and can be bought in stages.
You could start with the CMT 340's, then get the sub, then the rears. Fit the center in there when finances allow or save it for last, it matters not. The 340's should image well enough with just the front two until you get the center.
You could upgrade your receiver to a more full featured model sometime down the road, and the speakers have enough power handling capability to keep up.
Welcome to the forum.