1) JVC turntable. Turntables are tricky things, in that a dramatic difference can be heard based on the quality of the table itself and the needle. Remember, you are dealing with all analog at this point, and so there are several additional variables that need to be considered that you may not think of when dealing with digital. Is the turntable on a solid surface that doesn't receive vibration? Is the platter heavy, and turn evenly? Is the platter a direct drive or belt drive? How is the Tone Arm balanced, and is it balanced properly? What kind of cartridge/needle are you running? Are you running the correct preamp for that cartridge/needle? Is the speed calibrated properly? Are you using a dust cover? Are you keeping your records properly clean? Are the records flat (important when buying used).
2) The speakers. They are actually suppose to be good quality speakers. The one issue you may run across being older speakers is the condition of the woofers. As materials age, things dry out, and the woofer response is going to change over time. If they visually look OK, you are probably good to go, but just something to keep in mind.
3) Finally, the amp. Yes, a good quality amp will be a great benefit to speakers like that. Speakers that have giant woofers like that sound a LOT better when paired with an amp that has the ability to constantly supply a large current to properly drive that large woofer. A smaller, more peaky amp, may tire very quickly on sustained lower notes, whereas something with more power may be able to sustain it longer and precisely.
You don't need much power, depending on room size, and sitting distance from the speakers. Usually a good QUALITY 30 watt amp will do the trick. I can not stress the quality part enough though! Old Marantz receivers like the 2230 may only be a 30 watt/channel (60watts total) receiver, but it had a 220 watt power supply to sustain that over extended periods of time while driving large speakers. Looking at the back of the receiver to see what kind of power supply it has can tell a lot about a receiver, though this is usually relevant in class A/B amps. Class D amps are very efficient, and can sound great as well, but with a MUCH smaller footprint. I do find that they tend to lack the 'umph' to hit bass notes on large woofer stereo speakers as well as Class A/B amps though, but overall do a very good job for their size and power consumption.
As far as setup goes, it is really simple, and you have the right concept. Turntable --> Preamp --> Amp --> speakers. Many integrated amps have a Phono preamp for turntables built right into them, though preamps are often specific to MMC (Moving magnet cartridge) and MCC (moving coil cartridge). Unless it is an expensive turntable, chances are it is the common MM system, which virtually all integrated amps are made for. In all reality, with a good integrated amp/receiver, the preamp built into them is more than good enough, and may even be of higher quality than the one you are currently using.
In the end, those old speakers, when setup with the right amp, will may you want to throw the Aiwa system in the trash after it has been burned with fire. If you want to truly enjoy those records, you owe yourself finding a good quality amp to power those things. One of the best amps I can recommend to ANYONE for the money is the Marantz 2230 or 1060. They can be had for $200-300, and sound absolutely amazing. There really isn't anything 'new' you can buy that can compare for simple stereo listening in that price range, and they are just beautiful pieces of equipment. For new stuff, there are plenty of options out there, just do your research on "integrated amps". The good stuff usually starts at around the $400 mark.