Here's what I would go for with $700:
JVC SH99 MP3 Player - $230
Precision Power 4ch Amp (PCX-440) - $230
And depending on what kind of speakers your car can fit, I'd get
5.25 component speakers (infinity is cheap and sounds decent) - $100
6x9's (again, going with infinity) - $100
(you may have to drill out the holes for your tweeters)
Total is about $660 plus $40 extra for wiring (assuming you do the install yourself, which isn't very difficult). You will get SOME bass with this system, although it won't be felt outside the car. With this setup, the HU will run cool since your using an external amplifier. If you decide to add a subwoofer later on down the road, you can use your current amp to power the sub (and yes, it will power two 12's with ease because I have this amp and it is VERY NICE, CLEAN AND POWERFUL) and use the headunit's internal amp to power the speakers. But if your not a bass freak, the 6x9's should suffice. Its all about expanding later on when your not satisfied with what you have

and this setup will allow you to do just that.