If you're looking for a cheaper solution for replacement speakers, I highly recommend these sets.
They are astonishingly cheap, but I recently used them to replace the interior speakers in my girlfriend's Elantra, and even though I have an audiophile's ear, I must say that they really are not bad at all. Bass is solid without being too boomy, mids are present but never harsh, and the highs are crisp without being tinny. They could also reach pretty astonishing volume levels without distortion from the drivers - her head unit was likely the limiting factor, I imagine they could handle even more power.
I would highly recommend that, whatever you choose, make sure it's all 2-way. Most 3-way coaxial speakers sound like absolute crap because the midrange drivers are usually very cheap with marginally designed crossovers. This generally causes some significant buzzing from the mid drivers at high volumes. 2-ways are also much simpler to EQ than 3-ways, as a result of having the response curves of fewer drivers to take into account.
With regards to the head unit (HU) question, I would say it probably isn't worth the cost and hassle of replacing that at this time. The thing is, although most aftermarket HU makers will claim something like "50w RMS/Channel!!", they're usually lying. The Pioneer HU I bought recently was advertised that way, but further digging revealed it only actually puts out 12w/channel at 1% harmonic distortion. So that's pretty pathetic, really, and I paid $70 for it (luckily I only use it as a preamp). Aftermarket HU's have gotten wimpier over the past decade, and the manufactures more willing to lie about the specs, so the odds that you'll see significant improvement from upgrading the HU alone are pretty low. I'm not sure what the stock HU puts out, but I'm sure it's at least 12w/ch.
Whatever you do, don't buy a HU and leave the speakers stock. You'll see no improvement at all. My girlfriend's Elantra had an aftermarket HU with stock speakers and it sounded like dogshit. Upgrading the speakers is definitely your first priority, only upgrade the HU at this point if you really need features like USB, aux in, HD Radio, etc. And then be prepared to drop a c-note on a decent one.
Upgrades down the line: if you really want to push more power through the interior speakers, consider a 4-channel or 5-channel (if adding a sub) amplifier. There is definitely a limit to how much power you can get out of the amp in a HU simply because of their diminutive size. If you want to push beyond that boundary, you'll need an external amplifier. These things can be pretty complex to install, but if greatly improved output and sound quality is what you're after, it's worth it.
This would be a fine choice for an entry-level 5-channel amp:
So, as I see it, this is the order in which you should upgrade stereo components:
1. Interior Speakers
2. Head Unit
3. External Amp and Sub