skip the amp, that is for more hardcore audiphiles (especially if you don't know how to set one up yourself). For $200 including installation? hmm, I suggest you browse some sites and figure out how to hook speakers and amps up yourself, you'll save yourself hundreds of dollars in the future (and its a really great hobby).
It sounds like you need new speakers all around, and your JVC deck puts out decent wattage I'm sure (even 35x4 would suffice, although 40+ is much better) so it all comes down to what size speakers you have. Your fronts are probably 6.5" for 5.25"s if they're in the doors, and I'm not sure about your rears (either 5x7s or 6x9s). If you wanted more bass without upgrading to an amp and sub I highly reccomend getting some quality 6x9"s, they're great for bass. For a brand, I 100% always reccomend Eclipse...absolutely amazing equipment. You can probably get a pair of 6.5s or 5.25s for like $80 or less on eBay, and 6x9's will be right around $100 (plus or minus a few). If you can afford them, go with Eclipse. Remember to always try to stay with your car's stock size becuase if not, you're gonna pay hella installation fees for a custom job to resize your speaker holes.
My recs:
Eclipse if you can afford it, if not, buy Kicker (2nd best quality for the price, and also amazing sound)
Stay with stock speaker sizes (in your Elantra, prolly 5.25 and 6x9 it looks like)
6x9's provide great bass
Learn to setup speakes yourself, its really easy, especially when just removing the old speakers and hooking up the new wires.
If you have any questions, feel free to PM me, I'd be glad to help ya out. Good luck!