Here we go again...Autumn seems to be the time that everyone wants to upgrade their stereos.
Those hide-away subwoofers are a waste of money. They don't sound much better than your average stock subwoofer setup. They should fit under the front passenger seat but again, I don't recommend them.
First off...
1) Figure out your total budget you want to spend and let us know so we can help you pick out equipment
2) Decide what you want to get out of your system...more bass, better sq, mp3 playback, dvd player, xbox, etc...
Since I've already got an idea of what you are looking for I'll suggest the following setup...
* A 4-channel amplifier 50-75w x 4
* A set of component speakers (5x7", 6x8", or 6.5" will fit in The Mazda3)
* A single 8" or 10" subwoofer in a sealed enclosure
* A LOC (line-out converter) and 8ga wiring kit.
You can power the speakers with the front two channels of the amp and power the subwoofer with the rear two channels. You can mount the amplifier under the front seat and use quick disconnects for the subwoofer box which will go in the trunk. A box for a single 8" woofer will be about the size of two shoe boxes...but if you need the extra space, you can always remove it (hence the quick disconnects). Don't bother with the rear speakers, you won't need them...but for good measure you can continue to power them off the factory head unit.
We can help you pick out the amp and subwoofer but for the speakers, you should make a trip to a few of local retail stores and listen to speakers. This will help you decide what type of speaker your ear likes. Price will be a factor but if you can't tell the difference between a cheap pair and an expensive pair...there is no need to spend the extra money.