First you should find out what the impedance is of the existing speakers in the Klipsh ProMedia 2.1 system. You can damage an amplifier if you run speakers that are a lower impedance than the amp is designed to be stable at, e.g., 4 ohm speakers with an 8 ohm stable stereo amplifier.
In any event, you probably won't notice much of a difference between car speakers and the speakers already in the setup. In fact, the speakers already in the setup will probably sound better given that they are in an enclosure that was specifically designed for them.
MB Quart is generally considered to make the best coaxial car speakers, but they tend to want a lot of power to drive them, or else they won't sound so great. A 100W x 2 @ 4 ohms amp would be good for a good set of MB Quart coaxials.
For better sound, you'd want component speakers, that is, a pair of [usually 6.5"] midrange drivers and a pair of separate tweeters, with a passive 2-way crossover (usually 12 or 24 dB per octave). But again, you need a lot of power for quality component speakers, with 100W x 2 @ 4 ohms being minimum (I prefer 400 to 500W x 2 @ 4ohms for a pair of quality component speakers).
I'm assuming the Klipsh amp can provide 4ohms (although I couldn't find it mentioned in the specs) and this would be good for any 4ohm car speakers??
It isn't a matter of the amp "providing" 4 ohms; it is a matter of the amp "seeing" 4 ohms. You can hook any amp up to 4 ohm speakers, and it will work, but some amps (typical home audio amps for example, which are usually designed for an 8 ohm load) won't like it -- it sends more current through them than they are designed to be stable at, and if the amp doesn't have a protection/limiting circuit built in, it could damage the amp sooner or later.
Anyway, PC speaker manufacturers are in the business of making their stuff sound good with limited space and power in mind; a lot of engineering/experience goes into this. If you can randomly swap out a component of the system such as the speakers and have it end up sounding better than what you already have, it will be freak luck. It would be easy to start from scratch and make it sound better though, just throw a lot of money at it. Or you could make it sound better by starting from scratch and doing a lot of research, which should result in having to throw far less money at it, but a lot more time. Or you could get lucky with the off-the-shelf coaxial car speakers.