They can be cheap to really expensive. And yeah, I'll risk the downvotes, but a pre-amp isn't always necessary (in a pinch, I've had a turntable connected to some powered speakers, and it worked "okay"), but highly recommended you get one.

The signal coming from your turntable is weak. To put it simply, a pre-amp adds power to it. Also, pre-amps will alter the signal (what /u/38-RPM was referring to by RIAA equalization).

In a typical setup you have your turntable connected to a pre-amp, and you pre-amp connected to an amplifier (or a stereo receiver, basically the same thing), and your amp/receiver is connected to, and powers your speakers.

So: Turntable -> Pre-Amp -> Amplifier/Receiver -> Passive Speakers.

So, in your setup where you are just connecting it to your PC, you could connect your turntable to a pre-amp, and have your rca cable/headphone jack adapter going from the pre-amp to your PC. Just be sure to turn down the volume, since the new signal will be a lot more powerful.

I'll quite often see people mention the Art DJPRE II as the cheapest pre-amp they would recommend, but for your needs, something like a Pyle 999 will probably be fine. At the same time, there are pre-amps with USB out that might be worthwhile to look into for your needs.

Now, for grounding. Typically a lot of turntables have a grounding wire attached to their body somewhere, and you attach this to something such as a grounding screw on an amplifier/receiver.

There are ways around this, such as loosening a screw on the body of your amp/rec and attaching it that way, and some people would recommend sanding away any paint to create a metal contact point. The same can be done on your pre-amp if it doesn't have a screw point.

Now, I don't know enough about the following, so take the rest of this with a grain of salt. If your turntable is missing a ground cable, see if it has a spot for one, and see if you can replace it. If it doesn't appear to be missing anything, see what happens if you ground your pre-amp.