How to increase volume of my turntable - pre-amp - amp setup

at this point of time, get descent phonostage first and see the effect. than you can shop for descent cartridge.
needledoctor has now deal on 2m Red with headshell $129

Thanks for your help folks.

I think I understand that if I get a a phono preamp unit that is compatible with the type of magnetic cartridge in the turntable then all preamps will increase the signal to the same line level. Or at least any slight line level differences would be difficult for the casual listener to notice. So a different preamp might improve signal quality but not increase its level.

So it seems that what I need is something like this; https://www.amazon.com/TC-780LC-Stereo-Booster-BLACK-VERSION/dp/B000RZXAHI This component will boost the signal from the preamp on its way to the  Logitech speaker system. Other people use it for this purpose.

To me it would make sense for some company to combine the function of this box and a phono preamp into one unit. Maybe someone manufactures a component like this but i haven't found it yet.

Am I thinking about this in the right way or am I off the mark?

No, you need a phono pre-amp to go between your turntable and your pre-amp. The output of a phono cartridge is very low relative to a "line-level" component like a CD player or tuner. The output of a phono cartridge requires special equalization, as well as gain, before it is input into a pre-amp.

You can get a phono pre-amp from Pro-Ject, NAD, Parasound, etc., for about $100 - 150. You can find cheaper on e-Bay or at Parts Express, but I don't know how well they will perform.

Once you have a phono pre-amp in the system, the output when playing records and CDs will be similar.

@reubent is correct on all points. The last thing you need is a cheap device that boosts your output signal while increasing and adding noise in your system. And you don’t need more gain from your amp to the speakers, you need gain from your TT cartridge to the preamp.

Once you have a phono pre-amp in the system, the output when playing records and CDs will be similar.

This statement says it all. The output level from a cartridge is VERY low.

Yes, you need a phono preamp otherwise known as a phonostage. Given that your other components are so inexpensive you might search E-bay for an older Radio Shack phonostage. They can be had for ~$20.  Audio Technica made one that was selling for ~$50, you might search for one of those. I don't remember model number. 

If you are willing to spend more, Schiit Audio sells on for $129 that is highly recommended at that price point. You can also check out Needledoctor online.